There's been a bit of hooning on the moon before (Apollo 16, see below), but there's never been anything involving more than one wheeled vehicle that could really be thought of as a 'race.' But that looks like it's about to change, with the possibility of a three-way rover race coming in 2016. FINALLY.
The potential race would happen as part of the Lunar XPrize — a $30 million (total) competition to land a privately-funded rover on the moon. One of the criteria needed to win the prize is for the rover to land on the moon and travel 500 meters while sending back video.
Of course, one of the biggest issues with a race on the moon is just getting to the moon, which is, as you'll recall, pretty far away. Space X has a rocket, the Falcon 9, that should be capable of making the trip, and so far two teams, Astrobiotic from the US and Hakuto from Japan will be sharing one rocket to take three rovers to the lunar surface.
Astrobiotic will be sending their rover, Andy, and Hakuto will be sending two rovers, Tetris and Moonraker to compete. Astrobiotic benefits by having Hakuto as a first real customer for their long-term plan of a lunar-delivery business (which includes piggybacking on their Griffin lander for that very important final stretch to the moon), and Hakuto gets to the moon.
Here's where it gets fun: since these three are all landing at the same time, there will essentially be a race to see which rover can cover that 500 meters first and take the big prize. There isn't really any stipulation as to where the rovers cover that 500 meters, but hopefully for the sake of excitement, they'll all take off on a parallel track, so we can watch the first 1/6 gravity drag race ever.
Speed is not normally much of a factor for lunar rovers — most tend to creep along at a glacial pace, in order to maximize the science and observation time and to mitigate the risks of any crashes or damage. Now that there actually is some incentive for speed, I wonder if any of these three rovers may be modified accordingly?
Will Andy get some lower gearing for a quicker take-off speed? Will Moonraker be upgraded with the long-rumored Lunar VTEC, which could kick in, yo? Will Tetris be sporting some fat slicks on its two wheels? It already looks sort of like a dragster. I can't wait to see how this plays out.