For boardgames, stretch goals should indeed not be about adding or changing mechanics; that's a red flag. Stretch goals should probably mostly be about better quality components. Wooden or plastic pieces instead of cardboard, metal pieces instead of wood or plastic, a nicer board, more art, that sort of thing.
Extra options can work if options are already a big part of the game, and these are options the designer seriously considered but left out because they're not essential and too expensive to include. More money could mean they now can afford the non-essential options they originally intended but couldn't afford. But completely new game mechanics are a massive red flag.
It's worth noting that stretch goals are the area where a lot of Kickstarter projects mess up. Their original idea was good, but it was too successful and they promised too ambitious stretch goals which were just in the idea stage and never got properly developed, and suddenly they find themselves having promised things they don't have time for. I think this is one of the primary reasons why successfully over-funded projects end up being late.