"An element of a functor is an attaching functor into the category of elements of the functor," is unacceptably confusing due to the fact that the category of elements of a functor does not in any sense consist of the elements of the functor (as you would describe them).
I'm not sure where the above quote is taken from but I agree it is confusing. Here is my argument in favour of the traditional name. As Bill says, an element of an object F in a category is generally any morphism A --> F into F. It just happens that in many categories F is determined by elements with a restricted class of domains A. In Set, we can restrict A to be terminal. In a presheaf category, we can restrict A to be representable. The objects of the category elF of elements of F are (up to isomorphism) elements A --> F with A representable. It is also conventional to name categories after their objects (although the Ehresmann convention of naming them after their morphisms is more precise). Hence elF is the category of elements of F. Regards, Ross