I'm an adult female who doesn't play online (much, at least) and doesn't play most first person shooters and I currently have a 120gb hard drive in my 360 and I will be upgrading soon to the 250gb because I want the added space.
As others have said, it depends on the games you'll be playing. If you think you'll be buying things in the Marketplace, I would definitely recommend the larger hard drive. If you will be playing Rock Band or Guitar Hero, there are lots of songs available in the Marketplace and those can take quite a bit of space. There are lots of arcade games you can download, plus movies. You can also install some games on disk to the hard drive, which makes them run faster. And some "game of the year" editions have additional content (like new maps or quests) that is included with the disks you buy but can only be installed onto the 250gb system and not the 4gb system.