I would also say battery replacement is best. While the OS would be the same, he would unlikely benefit from a new phone. The only exception is if his eye sight is going bad already, in which case maybe upgrading to a 13 or 14 Plus would be good for him due to the larger screen size.
Even going to another model like anything from the 12 to 16 with the 6.1” might improve his experience, but granted not as much as the Plus models. Not recommending the Pro Max models since he would really be unlikely to use any of the extra features like extra camera.
But if his eyesight is good and he is satisfied with his experience already, the only problem is it’s not lasting as long then maybe a new battery is all he needs. Perhaps also consider a third party battery case which could further extend the battery life.