Preliminary evidence (2 runs with an account set to 6 different primary eidolons of one element) suggests the friend eidolon priority isn't based on the eidolon's characteristics at all, but instead on the party letter they're assigned to. A gets priority over B, B over C, etc. I'll need one more data point to confirm this (I shifted the eidolon in group C to group B, and the eidolon in B to C, with group A as my active party). I've done my testing with all LV1, LB0 eidolons, so it's still possible that such parameters could take priority and I've simply blinded myself to them. Two additional data points (a limit break, and raising levels of an eidolon) would be required to rule this possibility out.