The highly simplified damage formula is:

Base ATK x (1+assault) x (1+elemental) x (1+special)

Character attack goes into the assault value, which is where weapons assault and pride values go as well. Elemental includes elemental advantages. The nature of the formula means that you always want to increase the type that is the lowest.

When you are starting out and you have no weapon skills, anything that gives you the highest % is best. When you start leveling your weapons grid, assault tends to be much higher than elemental, so elemental becomes much more valuable. In theory, special attack is the most valuable since you can't get that from weapons at all (yet), but there are often special conditions attached to making the special attack active, which make them a pain to use.

tl;dr - elemental is usually the best for eidolons.