The Stripe jobs page (http://stripe.com/jobs) has some programming challenges that are way too ambitious for a usual couple-hour coding challenge, but definitely along the lines of a nontrivial, self-directed project that I'd like to see. Spend some time over a weekend making something like that, and it'll be worth noting to everyone you apply to. I've also seen great websites explaining mathematical concepts in intuitive, interactive ways; maybe there's a paper or something you like that you can turn into an instructional webapp.
Since you've got a couple of years before you'll be applying, another great option is to get involved with some open-source software, preferably something you use already or would want to use.
From what I've heard so far, it sounds like there is definitely opportunities for me, and that means that I can justify spending time working on these types of projects over the next couple of years.
Your above mentioned project is aligned to a new product we are gearing up to develop. I would love to chat with you further, you can email me at firstname.lastname@example.org