I understand you requirement, my father has an onkyo equivalent to the system that simonlewis suggested, and really struggles with it at times. He's 82 and has a mobile phone like the one in the link above. Still can't use it. So frustrating for him, and me.
So the one chebby suggested looks a good bet; my dad has a similar one in the kitchen and has no problem with it. I review these sort of things for a small publication that a friend runs. I can't think of anything new that is a) near your budget and b) even if it was would meet the usability requirement. Basically most new kit ditches the CD player for a iPod dock or for wireless, as that's what sells.
A slightly left field suggestion would be a small portable CD player (the round ones from Sony and the like) - have a look on Amazon. Linked to a Tivoli Model One bought on eBay. Slightly over yor budget but not by much. No presets, which might be a problem. But it does have a lovely tuning dial that your friend would probably be familiar with using. Especially if he/she listens mainly to radios 2,3,4, which are all near each other on the dial. The only issue, and it might be one, is that you have to plug the CD player in every time you want to use it. You can't switch to it from the front of the machine. Other than that it would be a good solution and sound nice as well. Maybe find someone with one and have a look.
Good luck