If you were to design a digital/board game, what elements would you care about the most?

Meaningful choice.

Every choice a player can make should have the potential for meaningful ramifications in the game. If a player can make a choice that has no meaningful ramification, it should be excluded from the game.

Is the idea fun? Would I want to spend however much time playing that game or not?

Yep, core mechanics should be simple and optimized I think.

I prefer games that emphasize skill over luck (few random elements) and have a lot of variation. One of my current favorites is Puerto Rico.

playability

Especially for board games this means simple rules that you can learn the first time you play, and the ability to have a different outcome based on your own choices. To allow the players some strategic options which will be balanced either by a different set of strategic options by the opponent (as in chess) or some extra luck (as in backgammon). Monopoly is a good example, where luck can make you avoid places where you have to pay a lot, but eventually if someone owns more than half the board it’s only a matter of time before they win.

I’ve designed a few games based on simple ideas like that. One of them has become a favourite among friends (basically because it allows different ways of winning it).