A single page site for people to quickly sell things. Craigslist, but faster.
Think sub 2 hour turnaround. Maybe simple sorting methods. Infinite scroll.
That's something that I could see being built in 2 weeks. Has the potential to become more than just a project too.
You could make a start page. Users can add calendar info, schedules, pictures of family/friends, and anything else that they want to see when they open up their browsers.
Of course this would also be useful for yourself if it's more of a personal project.
The usual advice is to build something meaningful or useful for yourself. (Track time, fitness, recipes, baseball, build a blogging tool, etc.)
Once you're comfortable with RoR you might branch out and build a Ruby gem or two, as these are relatively self-contained and allow you to focus on the language, testing principles, etc. (Some might suggest doing that first to become more comfortable with the language itself, but I think staying within the established bounds of the framework is a good way to get started.)
Gems, and an accompanying GitHub profile, are also good for the resume if you intend to seek employment.