I plan to introduce demon-bound magic items. In principle, I want them to be also accessible for player wizards to create, so I need to determine their rarity for crafting cost purposes.
For context, in game in addition to the crafting cost and time for preparing the item as an receptable, the caster would need to bind the demon into the item with Planar Binding, with the item replacing the 1,000 gp material component, and instead of being consumed, becoming enchanted. The duration could alternatively be set by the terms of a contract with the demon instead of the spell level of Planar Binding. To avoid bookkeeping, such a duration would either be defined as a specific event that is likely enough to happen so the enchantment is not perpetual, or a a daily or event driven probability, as in the example below.
Helm of the Screaming Demon
Wondrous item (rarity?), requires attunement
This brass helmet looks like the head of a demonic vulture. While wearing it and being attuned to it you have resistance to fire, cold, and lightning damage and can use Stunning Screech as an Action once per day.
Stunning Screech. The helmet emits a horrific screech. Each creature within 20 feet of it that can hear it and that isn't a demon must succeed on a DC 14 Constitution saving throw or be stunned until the end of your next turn.
Sentience. The Helm of the Screaming Demon is a sentient chaotic evil item with an Intelligence of 8, a Wisdom of 13, and a Charisma of 8. It has hearing and darkvision out to a range of 120 feet. The item can speak Abyssal, and can communicate with its wielder telepathically. Its voice is high and rasping.
Personality. The item is dull-witted, capricious and exists only to create pain and carnage. Its purpose is to make its wielder kill as many creatures as possible, and it tries to influence the wielder to do so. If the wielder has not killed a sentient being in three days or more, a conflict between it and its wielder occurs at the next sunset.
Bound Demon. Every time the wielder kills a sentient creature, there is a 1 in 20 chance that the terms of the contract with the demon have been fulfilled and the demon bound within is freed and returns to the Abyss. If some effect prevents the return, the Vrock demon appears in an unoccupied space next to the wielder, and tries to kill everyone nearby. The item loses all its magical properties (although it can again serve to bind another such demon).
My own thoughts so far: a ring of resistance which also provides protection would already be a rare item, and the DMG advises that combining the effect of several items at one rarity could bump it up to the next. But that ring is likely only rare because it could provide resistance to rare damage types. The merely uncommon ring of warmth for example provides resistance to fire and other benefits. The stunning effect is powerful but short, and by itself probably also should likely only be rare, a robe of scintillating colors is very rare, but the stun effect can be used three times as often. Taking all this together, my hunch would be to make this very rare.
I do not think that the limited amount of uses before the item loses its power is a big factor, because if the players can already craft this, the cost of re-binding another demon is rather small. If they could not, they might have to find the services of a powerful wizard for it, or quest to obtain a copy of summon greater demon, magic circle and planar binding, but here I am mostly concerned with rarity for determining the cost to craft, so they already would have those.
Why I find this hard to grade, is how to factor in the downside of this being possessed by a chaotic evil demon that will constantly try to influence and corrupt you (assuming the general attitude of the PC group is good-leaning). The other sentient item examples are all much stronger in ego and effects, and legendary.