Here are some ideas.
1. With a free account they get an email. If they upgrade to paid they get a text message (you can use Twilio).
2. You can partner with area hotels and limo companies. If the flight is delayed the email suggests hotels they can stay at or cabs they can take.
3. There are two parties involved: the person who is flying and the people who are expecting him/her. Target them separately. The person who is flying might be interested in letting his relatives or friends know that he/she will be late so that they don’t have to rush to the airport to pick him up. So to auto-notify other people you could charge a small fee.
4. Offer a subscription based service for people who fly frequently. This way their family always knows if they are running late.
5. Gamify it. When a flight is delayed people guess how long the delay is. Sell the guesses as options that are traded on a free-market (for points not real money). Use crowdsourcing to predict when the flight will actually arrive.
6. Calculate statistics. How often is a specific airline late? How often is a specific airline late to a specific airport on a specific flight? There might be other factors. Keep running scores on airlines. Rank them. People buying tickets could use the list to decide which flight to take.
7. Turn the list from #6 into a press release. Newspapers might run with it. Or it could make for a good blog post. Publicize it on HN. Create marketing buzz for your website cheaply this way.