Determining which fossil to choose from the Backpacker in the Relic Castle depends on which type of Pokémon the player wants. The Cover Fossil can be taken to the Nacrene Museum and regenerated into Tirtouga, a Water-and-Rock type, and the Plume Fossil into Archen, a Rock-and-Flying type.
While personal preference ultimately take control of this decision, there are some facts worth considering. Rock-and-Water types have more weaknesses than Rock-and-Flying types. Furthermore, Archen's learned moveset yields a wider range of attack types, with higher power ratings, than that of Tirtouga. Therefore, from a battling perspective, it would seem the Plume fossil would be a better choice.