I want my friend to be able to use a greatsword for his kensei weapon. Preferably I'd like to just add the ability to pick greatswords as a monk/kensei weapon but I don't see any way to add that to the choices. What's the best way to fix this?

Best way to do it would be :
1) Use the Tasha's Cauldron of Everything Dedicated Weapon Feature

2) Give him a feat that makes him proficient in Greatswords.

He should then be able to pick a greatsword as his dedicated weapon and everything should work.