The subject more or less says it all. Can I give my Lantern Archon a touch spell to hold, have it use it's greater teleport SLA, and have it deliver said touch spell? The deliver touch spells section states that the spell is only lost if the master casts a new spell. It does not mention whether or not a familiar could cast a spell.
Now, I would argue a SLA that is a touch spell might interfere simply because one touch should overwrite the other, but if the SLA isn't a touch spell, I don't really see why it wouldn't work. I'm not too fussed either way, but I thought I'd get the forum opinion on the matter.
The familiar casting a spell wouldn't be a problem (RAW) even if it could. But using a spell-like ability isn't casting a spell anyway, so you're on pretty solid ground.
OTOH, I too would like to know where you're getting a lantern archon familiar. AFAIK only harbinger archons can be familiar, and those can't teleport.
I suspect he is using materials from 3.5. A quick search suggests the Celestial Familiar feat from the Book of Exalted Deeds or the Planar Familiar feat from the Planar Handbook.
You're basically in uncharted territory, Pathfinder-wise. Paizo has consistently not made improved familiars that get teleport at will. Because the OP's issue should never come up, there is also no need for a rule about it.
There are improved familiars with dimension door (like the Silvanshee Agathion), which is the same thing for this purpose. Plus any SLA of any familiar really raises the posed question.