I am starting a small, computer system building business. The specifics, I believe, has potential to do very well (though I don't expect to become the next Michael Dell). The problem is, I have no money to my name to advertise with. Is there a good way to market that costs nothing (or, at very least, a minimal amount)?

The best free advertising is a good reputation and word of mouth, starting with your own. Always carry a lots of business cards, making it a point to give x amount (lots of them) away every week, do this while you tell exactly what can be expect from your service and of you, offer some kind of guarantee or follow-up service, Then when you are called upon, work to build a great reputation by being super professional: show up early, have a great attitude and show your gratitude for the opportunity to do business.

Give referral 'rewards' to satisfied customers.

Do Community service: Offer a free service or 'service donation' to a local community non-profit organisation with media clout and good local network with businesses such as a community radio station, library, schools, churches, etc.