The Mayor's assistant is assistant of The Mayor. She is regularly seen whispering into the Mayor's ear on multiple occasions, assisting him when he is confused. Although she speaks, what she says to him is not audible, though the gist can be discerned from how the Mayor reacts to her.
Her name may be Jen. Near the end of "Banned on the Run", after the Mayor is given a squishy bunny by Mr. Big, he looks to his right and says, "Hey Jen, come here, you gotta try this!" Although it isn't obvious that he was referring to her, he has never been seen working with other females at City Hall (other than Sally Botsford of course), so it seems likely that he was calling for her.