The general case of eigen-things has no other term to describe it. One might want something that hints at the geometrical meaning of eigen{values,vectors} when

The general case of eigen-things has no other term to describe it. One might want something that hints at the geometrical meaning of eigen{values,vectors} when they learn about them in the context of multiplying vectors by matrices, but ultimately the concept is a mainly algebraic one. Accepting this it becomes easy to deduce what should be meant by "eigenfunction of the differential operator". So I think, somewhat retroactively, "eigen" is the best word to use there.