If you don't want a nautical look, then stay away from red (which is my favorite combination!)
To add a nice bright accent to navy and white, I'd add something in a very bright, saturated hue, like lime green, sunflower yellow, fuschia, or tangerine. I'd stay away from brown and neutrals--they won't give you any splash. You don't need a lot to give the room some "oomph"! Put it on pillows, a throw, artwork, flowers, and trim on your curtains. Spread it around the room so your eye keeps moving.

You might consider a fresh, lime green.