Game Breaker: Any number over the years; see Loads and Loads of Rules. Most infamous, though, was the "operative" mechanic—rolled out, with insufficient playtesting (having been a last-minute addition to a set), just before that year's world tournament; discovered to be completely broken at said tournament (the hard way); errata'd within two months.
Scrappy Mechanic: Jedi Tests are rarely used, and certainly not enough to make a theme deck out of them. Also, TIEs with a hyperdrive (Vader's Custom TIE, TIE Defender, TIE Avenger) are generally better-liked than TIEs without one. Also the racial bonuses of Jabba's Palace never really got off the ground.