Sniper Alley is one of the "Spartan Ops" in the "Infinity" multiplayer mode. It can be played single player, split screen, or online with friends or other folks on Xbox Live. (For Xbox Live play, you'll need Xbox Live Gold, which is a monthly/yearly subscription that costs extra.)
Spartan Ops (and the other multiplayer modes) have to be installed, either from Disc 2 or from a free download. You don't have to play the campaign at all to play the Spartan Ops, although you might want to stick to offline play on the Easy difficulty if you're new to the game.
However, it sounds like you don't have enough hard drive space to install the Spartan Ops content. If you can, delete some things off the drive. You can do this from the dashboard or the Guide by going to the System Settings page.
There's a guide on this page which walks you through the process of locating things that can be deleted from your Xbox 360 hard drive. Scroll down to "Delete Games from your Hard Drive."
According to this article from Microsoft, installing Spartan Ops requires 1.45GB of storage. Do note that they've also released a second batch of Spartan Op episodes that weren't included on the disc, and are a separate free download.
It's also possible that you don't have a hard drive or that it is too small to contain everything you want to keep on the Xbox 360 - your saved games, installed games from Xbox Live Arcade, DLC, and so forth are all kept on this device. If that's the case, you'll need to buy a (larger) hard drive. What hard drive you buy depends on which Xbox 360 you've got.
Finally, do note that you can play the campaign cooperatively, so if you're just itching for some co-op time with your son, you don't have to go the Spartan Ops route.