I'm surprised no one is thinking of the car that won Motor Trend's Car of the Year honor in 1971 and game shows – from "Hollywood Squares," the 1970s "Name That Tune," "Let's Make a Deal," possibly early "Wheel Of Fortune" (there is a YouTube clip of a 1975 episode of "Jeopardy!" that advertised it as a prize) and, of course, TPiR – gave away by the thousands.
The car had incredible promise in its first year or two but then the reliability issues started coming and they definitely didn't get much better until the mid-1970s. It took replacing the engine itself with a reliable cast-iron four-cylinder engine to make the car at least somewhat reliable, but by then it was too late.
Of course, I'm speaking of the Chevrolet Vega.
Probably second on the list was a car given away not nearly so much except maybe very early in its history, but was also rife with serious engineering and reliability issues: the Chevrolet Citation. (Its General Motor siblings, including the Buick Skylark and Pontiac Phoenix, weren't nearly as hard hit but also didn't exactly escape recalls.)
Some might also categorize the 1976 Dodge Aspen/Plymouth Volare, which also had serious recall issues early in their lives, to the point where former Chrysler customers were abandoning ship and the company teetered on closure if fixes didn't happen. A couple of 1976 LMaD shows had this car as a prize.
I'd not be surprised if LMaD offered the 1969 Subaru 360 as a prize. Consumer Reports magazine once emblazoned its "Not Acceptable" mark of shame on the car for overall being inadequate (most notably being grossly underpowered and its crashworthiness). CR also gave the dubious distinction to the Suzuki Samuari (in 1988, due to handling issues that made it a serious rollover danger), a car that both "Card Sharks" and "Classic Concentration" offered as prizes.
Those are a few that come to mind.
Brian