Well, first, the sentence needs to be grammatically correct in that you would put Gregg first. As a general rule, "me" comes last. It's not I and Gregg, it's Greg and I....so first it would be “I so appreciate you taking Gregg and ___ child to school today.”
That being said, in this sentence, it should be "mine's".

“I so appreciate you taking Gregg and mine's child to school today.”

Mine is used for a possessive pronoun (your child) and my is for a possessive adjective.
You could also always say "Gregg and I so appreciate you taking our child to school today."