Regarding Isaiah 9:5(6), which text has the original rendering, LXX or MT, and why does the false rendering differ significantly from the original?

I would argue that, as a general rule, the Hebrew Masoretic text (MT) of the OT is more reliable than the Greek Septuagint (LXX) for the following reasons:

1. The LXX is a translation from the Hebrew, not the other way around
2. In numerous places, the LXX uses paraphrases and distinct biases
3. There are almost countless places where the LXX appears to use an unsupported original Hebrew when translating
4. The LXX is far from monolithic and uniform - several distinct sets of texts exist from ancient times
5. The MT is supported by the dead sea scroll text and shows negligible variation

There is another debate between Jewish translators and Christian translators about the four double titles applied to Messiah but that is another question.

First, we should note that the Dead Sea Scrolls version of Isaiah confirms the Masoretic reading, although this still does not put us earlier than the LXX. However, this fact does narrow the distance between the LXX and the precursor of the MT by as much as a millennium.

But more important, Jewish translators do not accept the typical Christian translation of this verse. One Jewish rendering of the text indicates that, rather than the child being given such titles as "mighty God" and "everlasting Father," these titles are ascribed to God, who names him simply "Prince of Peace."

For a child has been born to us, a son given to us, and the authority is upon his shoulder, and the wondrous adviser, the mighty God, the everlasting Father, called his name, “the prince of peace.”

Another Jewish version handles the translation this way:

For a child has been born to us, A son has been given us. And authority has settled on his shoulders. He has been named “The Mighty God is planning grace; the Eternal Father, a peaceable ruler."[Pele-yo῾eż-el-gibbor-avi-῾ad-sar-shalom.] (JPS)

(BTW this verse is Isaiah 9:5 in Jewish Bibles and some Catholic versions.)

Conclusion: Regarding the OP's main question, the LXX is not the original, because the original was written in Hebrew. Since the Masoretic and the DSS versions agree, they are more likely to be closer to the original. However, the Christian translation of the Hebrew tends to support a Christian understanding of the Messiah as a divine being, while the Jewish translations tend to support the idea that Messiah is a human being.

Correction

Note that the verse actually Isaiah 9:5 (in the non-latin versions and the original MT)

The state of the text

• ”כִּי־יֶ֣לֶד יֻלַּד־לָ֗נוּ בֵּ֚ן נִתַּן־לָ֔נוּ וַתְּהִ֥י הַמִּשְׂרָ֖ה עַל־שִׁכְמ֑וֹ וַיִּקְרָ֨א שְׁמ֜וֹ פֶּ֠לֶא יוֹעֵץ֙ אֵ֣ל גִּבּ֔וֹר אֲבִיעַ֖ד שַׂר־שָׁלֽוֹם“ (Isaiah 9:5 BHS-T)

As noted, the DSS confirms the Masoretic Text

• “ὅτι παιδίον ἐγεννήθη ἡμῖν, υἱὸς ἐδόθη ἡμῖν, οὗ ἡ ἀρχὴ ἐγενήθη ἐπὶ τοῦ ὤμου αὐτοῦ, καὶ καλεῖται τὸ ὄνομα αὐτοῦ Μεγάλης βουλῆς ἄγγελος· ἄξω γὰρ εἰρήνην ἐπὶ τοὺς ἄρχοντας καὶ ὑγείαν αὐτῷ.” (Isaiah 9:6 LXXS-T)

As the OP notes, the LXX goes its own way. We'll talk more about this further on.

• “Quia PARVULUS NATUS est nobis, filius datus est nobis, cuius principatus factus est super humerum eius, et vocatur nomen eius, Magni consilii nuncius, adducam enim pacem super principes, et sanitatem eius.” (Isaiah 9:6 V-LATINA)

As is its custom, the VL follows the LXX.

• ”ܡܛܠ ܕܝܠ݂ܕܐ ܐܬܝ݂ܠܕ ܠܢ݂ ܘܒܪܐ ܐܬܝ݂ܗܒ ܠܢ. ܘܗܘ݂ܐ ܫܘܠܛܢܗ݂ ܥܠ ܟܬܦܗ. ܘܐܬ݂ܩܪܝ ܫܡܗ݁ ܕܘܡܪܐ݂ ܘܡ݁ܠܘܟܐ. ܐܠܗܐ݁ ܓܢܒܪܐ ܕܥ̈ܠܡܐ. ܫܠܝܛܐ݁ ܕܫܠܡ݁ܐ.“ (Isaiah 9:5 PESHOT-T)

• Translation: "For it will be that an offspring will be born to us, a son will be given to us, and there will be dominion on his shoulders, and it will be that his name will be called wonder, and advisor, and mighty God, and forever will be his ruling of peace."

We note that that Peshitta conforms closely to the MT

• “parvulus enim natus est nobis filius datus est nobis et factus est principatus super umerum eius et vocabitur nomen eius Admirabilis consiliarius Deus fortis Pater futuri saeculi Princeps pacis” (Isaiah 9:6 VULG-T)

Not surprisingly, the Vulgate conforms to the MT.

The state of the LXX

We note that all of the versions except the LXX (and its daughter translation, the Vetus Latina) confirm the Hebrew reading.

Why is the LXX different here? This is one of the many places that show us that that LXX is a translation of the Hebrew. And as such, the MT is much, much, much more of a reliable source than the LXX. In rare examples the LXX is a better source if and only if there is evidence that the MT is questionable. Here there is no question at all. The other versions support the MT. And as such, the LXX is shown to be not-as-reliable.

A further example

This is not the only example of where the LXX is lacking. In my notes for Isaiah 25:8, I have the following:

The note here in the apparatus notes that in a few Medieval manuscripts the copulative is added. And in the Syriac it’s rendered as an Ithpeel (ܘܢܬܒܠܥ݂), which the editor concludes that the text could be repointed as a Pual in Hebrew. The result of the change would mean very little. Instead of “He” swallowing death, death “would be swallowed.” The editor makes note of it (evidently) because Paul, in 1 Cor. 15:54, (supposedly) goes with the text revised by Theodotian (Θ´ κατεποθη ο θ. εις νικος) instead of the standard LXX text (“κατέπιεν ὁ θάνατος ἰσχύσας” (Is. 25:8 LXXS-T)) The LXX here, in the standard text is nonsensical, making death the subject instead of “he.” It is possible that Paul is quoting בלע as a Pual (like the Syriac, ܘܢܬܒܠܥ݂). But it’s more likely that he’s quoting the MT more loosely to fit his audience. He’s definitely not quoting the LXX here. However, it may be the case that he’s quoting the Aramaic/Syriac, since he both uses a passive (ܘܢܬܒܠܥ݂), and he translates נֵצַח as ⲛⲓⲕⲟⲥ.

”ܘܢܬܒܠܥ݂ ܡܘܬܐ ܠܙ݁ܟܘ݂ ܠܥܠܡܝܢ“ (Isa. 25:8 PESHOT-T)

— And death will be swallowed up for victory forever

Pretty fancy: The Syriac gives us two words to cover BOTH meanings of נֵצַח.:

—ܙܳܟܽܘܬܳܐ victory; acquittal. —ܥܳܠܡܺܝܢܳܝܳܐ eternal, everlasting.

The translator(s) for the Isaiah section were not nearly as faithful to the text as they should have been. Earlier on in Isaiah 25, the translators give us a rendering that is incomprehensible. Putting the LXX at the same level as the MT is not tenable.

The Value of the LXX

The LXX is just a translation. It's a useful one. But it's not at all superior to the MT (again, unless, from a text-critical point of view, the reading in the LXX is supported by enough valid witnesses). Even in the NT, it is sometimes proposed that the LXX is what is quoted in the NT Greek. If one doesn't study the Greek very clearly, then I suppose one could arrive at that conclusion. But, whenever I find the OT texts cited in the NT, it's almost never word for word. It's partially LXX, but also then substantively changed to conform/refer to the MT original. As an example, the text I'm working on now is Hebrews 10. In that chapter, we read:

“ὁλοκαυτώματα καὶ περὶ ἁμαρτίας οὐκ εὐδόκησας·” (Ἑβραίους 10·6 THGNT-T)

In my notes I have the following written down:

LXX comparison:

Alexandrinus (& the NA/UBS text) has ⲟⲩⲕⲏⲩⲇⲟⲕⲏⲥⲁⲥ here, whereas the LXX has “οὐκ ᾔτησας.” (Psalms 39:7 LXXG-PS-ODES) This more flexible quotation of the psalms might very well be telling us that the writer to the Hebrews is making use of a Hebrew source, not a greek one. The masoretic text has: ‎לֹ֣א שָׁאָֽלְתָּ “You did not ask [for].” The “you were not pleased” rendering is perhaps just a more functional translation of the Hebrew than “you did not ask.”

Masoretic Text:

”זֶ֤בַח וּמִנְחָ֨ה ׀ לֹֽא־חָפַ֗צְתָּ אָ֭זְנַיִם כָּרִ֣יתָ לִּ֑י עוֹלָ֥ה וַ֝חֲטָאָ֗ה לֹ֣א שָׁאָֽלְתָּ׃“ (Psalm 40:7 BHS-T)

Versions:

LXX

“ὁλοκαύτωμα καὶ περὶ ἁμαρτίας οὐκ ᾔτησας.” (Psalms 39:7 LXXS-T)

Latin

“holocaustum et pro peccato non petisti” (Psalms 39:7 VULG-T)

(Note how Jerome stays close to the Hebrew Text)

“Holocausta etiam pro delicto non postulasti,” (Psalms 39:7 V-LATINA)

(Here too, the old Latin stays close to the Hebrew: “For whole burnt offerings for sin you did not ask for.”)

Peshitta

”.ܒܕܒܚ̈ܐ ܘܒܩܘܪ̈ܒܢܐ݂ ܠܐ ܨܒܝ݂ܬ“ (Psalm 40:7 PESHOT-T)

“With sacrifices and with offerings, you did not want (be pleased with).”

(The Peshitta conforms to the MT)

Even when the LXX is cited in the NT, it is done so with the MT as a base text.

Biblical Inerrancy Implications

The difference between the LXX and the MT in this verse does not affect biblical inerrancy at all. It simply shows us one of the times that the LXX isn't as faithful as we might have liked.

All four answers so far seems to agree that the MT version of Isaiah 9:6 is closer to the original. Whatever the original rendering was.

No one, however, triad to explain the logic behind the LXX rendering.

There is a link to an interview that explain the most likely reason:

https://www.joelmadasu.com/why-does-the-septuagint-say-mighty-counselor-of-god-in-isaiah-96-an-interview-with-dr-michael-heiser/

As I don't have the time to produce a transcript, I am only highlighting the most important points.

The main reason of the LXX rendering is early Jewish binitarianism:

• Language used in the verse (פֶלֶא H6382, יָעַץ H3289) is asociated with God (JHVH) but also the angel of the LORD:

O Lord, you are my God; I will exalt you; I will praise your name, for you have done wonderful things, plans formed of old, faithful and sure. (Isaiah 25:1)

This also comes from the Lord of hosts; he is wonderful in counsel and excellent in wisdom. (Isaiah 28:29)

therefore, behold, I will again do wonderful things with this people, with wonder upon wonder; and the wisdom of their wise men shall perish, and the discernment of their discerning men shall be hidden.” (Isaiah 29:14)

So I will stretch out my hand and strike Egypt with all the wonders that I will do in it; after that he will let you go. (Exodus 3:20)

“Who is like you, O Lord, among the gods? Who is like you, majestic in holiness, awesome in glorious deeds, doing wonders? (Exodus 15:11)

And the angel of the Lord said to him, “Why do you ask my name, seeing it is wonderful?” (Judges 13:18)

• The theology of the Angel of the LORD, who is different from JHVH and is JHVH at the same time.

2 And the angel of the Lord appeared to him in a flame of fire out of the midst of a bush. He looked, and behold, the bush was burning, yet it was not consumed. 3 And Moses said, “I will turn aside to see this great sight, why the bush is not burned.” 4 When the Lord saw that he turned aside to see, God called to him out of the bush, “Moses, Moses!” And he said, “Here I am.” 5 Then he said, “Do not come near; take your sandals off your feet, for the place on which you are standing is holy ground.” 6 And he said, “I am the God of your father, the God of Abraham, the God of Isaac, and the God of Jacob.” And Moses hid his face, for he was afraid to look at God. (Exodus 3)

11 Then the angel of God said to me in the dream, ‘Jacob,’ and I said, ‘Here I am!’ 12 And he said, ‘Lift up your eyes and see, all the goats that mate with the flock are striped, spotted, and mottled, for I have seen all that Laban is doing to you. 13 I am the God of Bethel, where you anointed a pillar and made a vow to me. Now arise, go out from this land and return to the land of your kindred.’” (Genesis 31)

15 Then he blessed Joseph and said: The God before whom my fathers Abraham and Isaac walked, the God who has been my shepherd all my life to this day, 16 the angel who has redeemed me from all harm may he bless these boys. (Genesis 48)

The Israelites were led to the promised land by the angel of the LORD according to Exodus 32:34:

Now go, lead the people to the place I spoke of, and my angel will go before you. However, when the time comes for me to punish, I will punish them for their sin.”

And they were led by God according to many other verses, like Deuteronomy 8:2:

Remember how the Lord your God led you all the way in the wilderness these forty years, to humble and test you in order to know what was in your heart, whether or not you would keep his commands.

Jude in the NT identified Jesus as the angel and JHVH:

4 For certain individuals whose condemnation was written about long ago have secretly slipped in among you. They are ungodly people, who pervert the grace of our God into a license for immorality and deny Jesus Christ our only Sovereign and Lord. 5 Though you already know all this, I want to remind you that the Lord at one time delivered his people out of Egypt, but later destroyed those who did not believe.

The best proof that Heiser is right, is another paraphrase of Isaiah 9:6. It was also produced in the intertestamental period by another group of Jewish theologians. This text, too, incorporates the early Jewish binitarianism. It is Targum Jonathan on Isaiah 9:

The prophet said to the house of David, For unto us a child is born, unto us a son is given, and he has taken the law upon himself to keep it. His name is called from before Him who is wonderful in counsel, the mighty God who liveth to eternity — the Messiah whose peace shall be great upon us in his days.

https://www.sefaria.org/Targum_Jonathan_on_Isaiah.9.5?lang=bi&with=all&lang2=en

So, the Messenger/Angel of great counsel is the perceivable manifestation of the unseen God, in the form of a newborn baby. Which is possibly more Christological than MT rendering.

In general, the Hebrew has many differences from the Septuagint; and I wonder why. There was probably a different text from what the Septuagint translated, or the Hebrew changed before the 1st century, because we see from the translations of the three Greek translators, Aquila, Symmachus, and Theodotion, between the beginning of the 2nd century (130 AD) and the beginning of the 3rd century (i.e., within the limit of one century), that they are in accordance with the current Hebrew. But many current translations are not as they should be from the Hebrew text.

The LXX translation is this (from the critical text of the Göttingen Academy of Sciences or the text of Rahlfs-Hanhart; on this verse, both are same):

For a child is born to us; a son is given to us; whose dominion is upon his shoulder, and his name is called the angel of a great plan; for I will bring peacefulness to the chiefs, peacefulness and spiritual health by him.

The translation from the Hebrew is approximately this (from the critical text of the BHS):

For a child is born to us; a son is given to us; and the dominion is upon his shoulder; and his name is called the true God, who plans wonderful thing(s), the timeless father, the lord of peacefulness.

The Masoretic text has the original rendering, and the usual English translation by Christians is also correct. However, the passage is misunderstood by Trinitarian exegesis.

Isaiah 9:5(6)b may be translated:

And he is called "A Wonderful Counsellor is the Mighty God, An Everlasting Father is the Prince of Peace."

But Isaiah 9:5(6)b is best translated:

And he is called Wonderful Counsellor, Mighty God, Everlasting Father, Prince of Peace.

All four epithets belong solely to God the Father, but are here attributed to the Son, because the Father is actually inside of the Son, and the Son is the direct agent of God as the mediator to men.

The Son is the exact Image of God, and therefore when one sees the Son, they indirectly see God the Father through him.

The Son is appropriately called the titles of the Father God because he is God's exact representative to men, and the Spirit of the Father himself resides within the Son.

Isaiah 9:5(6)b does not imply that Jesus is God, but that the Mighty God is in Christ, and God is manifest to us through Christ his Son.