Brake stop light switch, located in the vicinity of your brake pedal. It's about a $100 part believe it or not for your 04 Mountaineer. Before shelling out a C-note for it, pull it out...clean the terminals, and see if that helps. Although, it will be pretty apparent if it's a mechanical failure, if the button mechanism is jammed. It literally attaches to your brake pedal lever, and activates the brake lights...a simple little part, unfortunately it's quite pricey for these vehicles. I've seen them for less than 10 bucks in parts stores for other makes however.
Just looking at AutoZone, they want about $140 for their Duralast brand, but I did find it on rockauto.com for about $100...so shop around.