"That said... I doubt I would ever take the time to assign an alias to everyone I am following, so I am not sure how much use this would get in practice." That's why the client should make these entries at the time you are first following somebody. If I follow Dikaios1517 and you change your name tomorrow, I will have no idea who's that but I might remember who I did follow yesterday.
The client should be mindful of course. So I would like my client to show both pfps in some way. For example the old one on hover with a mark hinting at a change having happened. So when I hover the pfp, a second pfp would slide from under the current one, allowing me to pick one, after which the pfp would stop being marked until the user changes his pfp again.
That could be interesting. You can manually set an alias for them, if you want, and your client will automatically do so for you if they change their handle to something other than what it was when you first followed them.
I also like the "old PFP sliding in from the side on hover" for desktop clients. Some indicator to tap would probably work on mobile.