I think it's important first to make sure your offering is compelling. A contest is great, but only if what you have to offer is compelling enough to remember. I would propose you take that $100 and use it to test the idea you're working on with your target audience.
Before spending money on this (or other forms of paid marketing), you should have a reasonably accurate idea of what your sales and marketing funnel look like.
Basically, figure out the economics of your business. What's the conversion rate from website visitors to leads? Leads to customers? What's the revenue per customer?
Instead of a contest, you might be better off spending some money on Google AdWords. Even though this may not generate sales, it'll give you a better sense for the keywords that "work" and is much more scalable than contests.
There are entire online communities of people who as a hobby, spend their time entering their details to various online contests. You might see a ton of hits, but I really doubt you would get many conversions.
If you're asking for a technical way:
If you post it to twitter with a link, you can simply check the HTTP referrer header for visitors on your front page. Google Analytics will do this for you and sort and give you percentages of who came from where. This will help you figure out where people are coming from even if you don't have a big marketing scheme going on.