I do like this idea. I think we could use more things to keep us engaged during long sails. Games were something teased early on with Sea of Thieves. Such as Karnath, a game played using dice or cards.
A new interactive form of ship trinket would be good for games. I know a certain 'sit-on' barrel below deck on the sloop that would be perfect to layout a board game on. I'm sure there are similar spots on the larger ships, just can't think of them right now.
Any game where you take turns, like chess and backgammon, one player could be away watching the horizon and manning the helm while the other takes their turn. When that player's turn is over they can relieve their crewmate to go below and take their turn. Being Sea of Thieves, cheating could be a thing. Maybe you move some of your opponents pieces before taking your turn at helm.
Being trinkets it would be possible to limit the type of games you would have based ship crew size. Sloops you'd only have access to 2 player games while a galleon has access to 4 player. As mentioned earlier, being a trinket also means that the game could get jostled and have to be reset. Unless you "save" it. In the Heart of Fire novel they mention using wax to hold chess pieces in place between games. Perhaps something like that could be an option.
Another idea I have is a new type of equipment. Where you'd buy games from the merchants. The components of these games could be themed after the different trading companies. A crewmate would then pick which variant of these "pocket" games they wanted equipped as you would with any of your gear. When activated they'd lay it out wherever they are. Play on the deck, beach, even in a creepy cave. These would likely have to be dice and card based games. Something that can be quickly pocketed when trouble comes around.