From Curse of Strahd:
A dark gift functions like a charm (see “Supernatural Gifts” in chapter 7, “Treasure,” of the Dungeon Master’s Guide).
OK so looking at the DMG about Charms:
If a charm lets you cast a spell, you are able to do so without spending a spell slot or providing any components (verbal, somatic, or material). In any case, a charm can’t be used in the area created by an antimagic field or a similar effect, and a charm’s effects are susceptible to dispel magic and the like. But the charm itself can’t be removed from a creature by anything short of divine intervention or the wish spell.
Neat they don't need components, but doesn't help us.
Below I will give suggestions and their reasoning.
Use Any Existing Spellcasting Ability
I would fall back on the default magic item rule: Pick any of your existing spellcasting ability scores. This interpretation, while originally meant for specific magical items grants the player agency in its use. And means it should effective and easy to use.
A magic item, such as certain staffs, may require you to use your own spellcasting ability when you cast a spell from the item. If you have more than one spellcasting ability, you choose which one to use with the item. If you don't have a spellcasting ability - perhaps you're a rogue with the Use Magic Device feature - your spellcasting ability modifier is +0 for the item, and your proficiency bonus does apply.
Use Spell Scroll DC
Looking at charms in general, they often grant temporary or permanent access to a magic item, without the physical item. As such it would be logical to assume charms granting access to spells could be considered access to spell scrolls and the ability to use them
So it would be sensible to use the spell scroll DC for scroll that would allow access to that spell. It is also easy to use as you can just effectively give the player a number of spell scrolls of the ability (with the knowledge they are charms instead of physical scrolls).
Use Charisma Spellcasting Ability
Charisma is involved with the gifts already by their saves. It is also a statistic commonly associated with innate spellcasting. Further, its also used by warlocks, which channel powers gifted by other (non-divine) beings.
Use Constitution Spellcasting Ability
If you envision the gifts granted as enabling the PC to power the gifts through their own life force or energy, Constitution is also a valid ability score to associate with it, as it also is sometimes used with innate spellcasting.
Use No Spellcasting Ability
Very technically, this is likely the "correct" answer. If you are granted the ability to cast a spell, but not granted a spellcasting ability for that spell, the game actually has a method of resolving that situation in the DMG, already quoted, but here again in brief
If you don't have a spellcasting ability - perhaps you're a rogue with the Use Magic Device feature - your spellcasting ability modifier is +0 for the item, and your proficiency bonus does apply.
I cannot suggest this method as it decreases the effectiveness (and fun) of the gift, and is unlikely to be the intended effectiveness of the gift. But it is likely the "As Written" solution.