1. Get people to tell you when they're sick and what symptoms they have. 2. Get access to their social network (facebook app, email contact list, etc) and figure out who they got their cold from. 3. Tell users who got them sick and what they should expect (e.g., "looks like you picked up Joe's cold, but don't worry -- it's mild and you should recover by the weekend").
Possible future directions: 1. Given enough users, you could do a very good job of tracking influenza or other rapidly spreading infectious diseases. 2. Have people tell you what medicines they tried and what worked / didn't work (very useful in the context of antibiotics).
Monetize via advertising (good rates in the medical field) or by referrals to drug stores which sell cold medicine. Get publicity by publishing statistics (e.g., "looks like the flu season has started in California, but hasn't hit New England yet").