With the way synthesis works, I don't think there's any way it can (I don't think using Synthesis to Manifest allows for the teleport option, ie- the Eidolon has to be unmanifested to use synthesis).
I think effects like Mage Armor should transfer with Synthesis, but that isn't how it works at the moment.
If you cast mage armor on yourself does your Eidolon get +1 item bonus?
What if you used Synthesis?
Your Eidolon can cast it on themselves with Greater Magical Evolution as an innate spell at 8th.