As for friend-list folks equipping useful Eidolons, don't the choices you get go off of the Support Eidolons they put in the element tabs in their Profile, and not the ones they actually have equipped?

I noticed a spike in my daily "you've been chosen for support" summary after I equipped a Loot Drop Eidolon into the support tab for its element, but I don't actually use that Eidolon in my main party setups.