Grist is what you grind in a mill. Since the rumor mill is low on grist, that means there isn't much in the way of gossip these days. Grist to/for the mill also has the idiomatic meaning of "something that can be employed to someone's advantage" often when the something does not appear advantageous.
So the double meaning here is that there is not much gossip that she can exploit for her own ends. It really is a clever play on words.
The "give it to me" part I couldn't say. It could mean "tell me what little gossip you've heard" or it might mean "hand me that thing" or it might mean "keep up the witty reparté." There's really not enough context to tell.
EDIT: "Only by marriage" means that they are only related through marriage, not by "blood" or genetic ties. How this relates to their relationship, I don't know. Is he her step-father? In any case, she is teasing him; if they are only related by marriage then she didn't inherit her cleverness from him, which is what he is suggesting by saying "you must be related to me."