I love to do this, because often times I had students in my classroom that had NEVER played a board game! They loved it, and were so engaged that sometimes throughout the year I would pull them out again.
Some games that I liked to use:
1. Connect four
3. Tick tack toe
4. Candy land
7. Don't spill the beans
8. Monkeys in a can
12. Fishing game
15. Jenga (I also had the large set)
There are probably more that I am missing, and there are even other games that I put out all of the time. Usually those are more curriculum based and could be put together to match common core etc.
I challenge you to start out with 2 games, and let your students have fun, remember some may never play a board game again!