First, I will pile on to the “throwing good money after bad” train.
That said, if you want to improve the shifting, there are probably more important things to do before buying new components (this is assuming you have already tuned it as well as possible with the cable tension and limit screws):
1- Set up the cables and housing correctly. These bikes usually have The cables and housing cut WAAAAAY too long, which degrades performance (especially with cheap cables/housing)
2- check the hangar alignment. Preferably with a tool. If you don’t have one, consider buying one. It will do a lot more good than whatever money gets wasted on trying to upgrade this bike.
Otherwise, if he wants to spend money on this, do so with things he can use after he wisens up and gets a new bike, such as pedals and shoes. Or (and I know this sounds absurd on a BSO) a dropper post.