It is interesting to note that God is speaking in the past tense regarding what he has already done for king David. A different translation of the text in question continues past tense with "...and I have made thee a great name, like unto the name of the great [men] that [are] in the earth." (K.J.V.)
This makes those great ones either contemporary with David, and/or with those who had preceded him. Given that God is speaking of David ruling his people, and how God had cut off all his enemies, the simple reading is that there were other kings / rulers who had also achieved victory over enemies and become renowned for their greatness as rulers, from way back until the days of David.
Then God speaks of the future, after David will have died, with the house of David being set up by God. This speaks of David's descendants, Solomon already being promised by David to Bathsheba to take over his throne, as in this verse:
"And when thy days be fulfilled and thou shalt sleep with thy fathers,
I will set up thy seed after thee, which shall proceed out of thy
bowels, and I will establish his kingdom." 1 Samuel 7:12 K.J.V.
This is where it is interesting to note something about "greatness". In previous verses we learn that David conducted himself very wisely, so much so that the then-king Saul was afraid of him (1 Samuel 18:14-14). But when king Solomon took over the throne of David, his father, his wisdom was so great that it became proverbial. The queen of Sheba traveled a huge distance with an entourage and gifts of immense value to check Solomon out with regard to his renown wisdom, even back then. She had not heard the half of his greatness in that respect. But the wisdom of Solomon became so proverbial that the saying continues to this day, on a global scale.
So, greatness is not only about kingly power and battles, or the size of a kingdom. In the case of David it was very much about his spiritual greatness, and there were men of old, such as Melchizedek, Abraham, Noah etc who were spiritual giants of their day, and have gone down memorably in history. Whether secular kings and world powers up till David are included in the verse in question is not stated in the text. But there is enough that is stated to get a general idea.
That pen-ultimate paragraph answers the main question. As for the subsidiary one, "[Wasn't] David's name already great?", the simple answer is "Yes, for God spoke of having already made his name great, and into the future through the likes of Solomon, but supremely through "the Root of David" - Jesus Christ, the "greater than Solomon" (Matthew 12:42, Rev. 5:5, 3:7, 11:16). That name would long outlast all the other great names. For Christ has been given the name that is above every name, and every knee will bend to him (Philippians 2:9-10).