There are so many good games on the Lynx its hard to know where to start (though i would with
Shadow of the Beast 
).
In my experience though, while the Lynx is good fun in single player, the multiplayer ComLynx games is where the Lynx really starts to shine (and even topple the 'opposition'). Its even worth getting another Lynx and a comlynx cable if you can, along with the necessary comlynxable games. Some of the best multiplayer networked games for the Lynx:
Checkered Flag (1-6 players) - Great formula one racer. I don't normally like car games but i enjoy this one. Good on its own but best in multiplayer.
Robosquash (1-2 players) - 1st person Raquetball type game with power-ups. OK in single player, GREAT FUN in 2-Player!!
Slime World (1-8 players) - Great exploration game that gets better with each additional player.
Zarlor Mercenary (1-4 players) - one of my all time favourite SHMUPS! More challenging with 3-4 players but as a single player its rather FANTASTIC as well!
Just a word of warning: i would invest in a power supply if you haven't got one already - it will save you heaps of money (new batteries) and time (recharging batteries)!
Most of all, enjoy your Lynx