Considering that "age" is a property of living things (I mean, think about it . . . it's the distance in time since something came into being, which for the universe is pretty irrelevant, but for you and me is pretty freaking important), we'll have to assume that something would stay alive inside of a black hole. Which, considering the gravity, is impossible. But let's ignore that for a second. In a biological sense, "aging" is really the modification of certain processes. Your cell division slows, you accumulate mutations, your telomeres shorten, your skin and hair become less elastic and start disobeying laws of physics all on their own. So if something was still alive inside of a black hole (which it wouldn't be) and its biological processes were still carried out (which they wouldn't be), then I argue that they would still age.
Of course, none of this is remotely realistic, so I will stop. But go ahead and have that deep, Saturday night conversation about whether aging is dependent on time or just coincidental . . .