Dime, as in US 10c, so it refers to money/payment. "On our dime" means "at our expense" (ie. we are the ones being made to pay for you).
There's another phrase "it's your dime", used to encourage someone to simply speak directly. It's best explained in this telephone conversation:
A: Hi, can I ask you a question?
B: It's your dime. (=you're paying for this call, so just go ahead and say it.)