I think it'd be cute to have a little box in or above the companion box so you can nickname the companions! Or perhaps it changes the spot that says Companion into the nickname. It'd clear out when swapping them out and shouldn't affect the companion page.
This is a very quick mock up where it would be in the spot where it just says Companion normally. In this case I changed the nickname of the Onyx Dustraven to Murkrow.