For some fun, since he is that specific multi-class, you could just make available to your monk the possibility to get a pair of gloves with reinforced knuckles or some gauntlets to count as weapons that have the finesse property.
If a monk wears a pair, the damage from the gloves progress like the monk's unarmed attack (you could even start making +1, +2, +3, martial arts oriented gloves), and the wearer can´t be holding any other weapon in their hands, or maybe trying to attack with a weapon while wearing the gloves gives disadvantage to that attack.
If he ain't wearing them by any reason, he can't use Sneak Attack with his fists (note I didn't say "unarmed attack").
Another option could even be restricting this multiclass Monk/Sneak Attack choice to just be possible with those kind of gloves, and no other rogue usual weapon. Flavor!
If any other class wanted to equip a pair, they just works like clubs, no finesse.
From a coherence point of view, you can hit somebody with your bear hand in a precise and specific spot and knock them out, even kill them. Throat, temple, back ear, and so on. Everybody knows, more or less.
Add a well bandage wrapped protected hand in a glove with some metal, wood or bones in its knuckles, and things gets harsher for foes and more bearable for your hands.
Rogue's Sneak Attack feels like something very suitable: vicious surgical strikes with small tools against weak spots.
I hope some day they will finally add something like it officially, doesn't make much sense to me they haven't yet, the whole knowledge and concept about a martial artist class has evolved a lot from the narrow classic one, something totally understandable putting it in the correct context in its time frame, though.
I would say the Monk class really needs some love.
Edit: I apologize for not being able to answer the question clearly, I realized thanks to Longspeak. Trying to amend the fault, I would add to my first post:
You could make unarmed attack eligible for Sneak Attack but just for one try per turn, not the additional ones that can come after because martial arts and ki. If you miss your first attempt to make sneak damage, the enemy is aware of your devious moves and , even if you're not losing advantage, you can´t make it work for the moment (waiting until next round), and while on the heat of battle you have no seconds to lose and immediately take other course of action adapting yourself to the circumstances.
Hopes this suits better for a proper subjective answer.
P.S.:Thank you Longspeak for being so polite in your correction. I notice and appreciate those details. Nowadays even more, since internet is so full with bitterness.
Have a nice day!