To be firmly committed to a decision, preference, opinion, affiliation, etc. This phrase is actually most commonly used in the negative to imply one's flexibility on a subject or lack of staunchness for a particular decision.
dude 1: "Anyone have a preference were we go for dinner?"
dude 2: "Let's go to Wendy's. Not that I'm married to the idea, but a frosty sounds real good right now."
dude 1: "Wendy's it is!"