Once per short rest would almost certainly be too much. That means you might get as many or more castings of a 2nd level spell than a dedicated Wizard of your level (they get 3 at level 4, for all of their spells).
Once per long rest might be okay. This is more in line with Magic Initiate (gaining a slightly better spell once per day but losing out on the cantrips).
It's also not unheard of for non-casters to gain access to a 2nd level spell. Various races gain them as racial abilities at level 5. Some even get some (not
very powerful) 3rd level ones.
There's also a few items that let you cast 2nd level spells when not a caster, so also in that regard it's not overpowered.
Also in the specific case of giving Mirror Image to a Barbarian, it seems fine to me... the Barbarian is essentially already a massive punching bag, at best this'll block a few more points of damage at the cost of your action.
So I'd be fine with allowing this, although I'd wonder about "how" in this case. Lore-wise, are you actually learning this one specific spell, or is the ability coming from something else? But balance-wise that's not a consideration.
The only thing I'd maybe consider, especially if you generalize the Feat to "any spell" is to slap on a "character level 3+" restriction. It might be a bit much for a Variant Human to have a 2nd level spell at level 1. (Probably still not overpowered, but better safe than sorry if you're allowing this as a full option)