I went through the same thing with my dad. A lot of guys will discuss different cartridges. But different cartridges are not really the answer. The answer is in rifle design and rifle weight. Now the problem with weight is that it is difficult for an older shooter to handle a rifle if it is too heavy. So normally the weight needs to be controlled also. So this leaves rifle design as your first priority.

We went to a thumbhole stock Encore for my dad. The thumbhole design spreads the recoil out. It allows for your hand and arm to absorb part of the recoil.

Personally I do not like quarter bore rifles. But for my dad we went with the 257 Roberts. The 250 Savage would also be a good choice. We went with the medium weight barrel with a 20 inch length. This makes for a rifle that handles very well. Weight without scope is just under 8#. Then we put a 2X7 Leupold RF scope on the rifle with the shorter parallax, and sighted it dead on at 75 yards. My dad was deadly with that rifle. And 100 yards was farther than he could see effectively.

Find a rifle with a thumbhole stock somewhere and see if your dad likes the design. I know my dad really loved his. Tom.