Start with any industry/business you like. Find any business process which involves a telephone and a set of repetitive instructions involving it. Turn it into a Twilio script. Take that to business owners, office managers, etc. Demo the script for them. If their eyes light up and their checkbooks come out, you have a good idea.

To get you started you might want to implement a Google Voice like application. Though that sounds that might sound difficult, using Twilio it can easily be done in a weekend. It will teach a lot about how to create Twilio applications plus it gives you the freedom that Google Voice lacks.

For example I hated getting phone calls when I was in meetings. So I hooked my Twilio app into my Google Calendar API and bam, when I was in meeting Twilio would ask them to leave a message or press 5 if it was an emergency. Also greeting people by name when they call is pretty awesome.

There is a big list of them in Patrick's post, maybe that would be a good place to start.

There are tons of cool little ideas out there, and plenty of big ones as well.

I'll plug one of my recent creations: http://shouldbeback.com/

Twilio contests are a great place to see what interesting things people are building and the cool API Mashups that are possible http://twilio.com/contests/. The most recent was a Shopify+Twilio hack that had some really cool results http://bit.ly/vBrGRx

Write a power dialler, that is a system that takes a list of numbers and the number of agents you have available and will ring enough customers so that the agents are on the phone more often than not.

Sell this to people running call centres, make sure it stays up all the time and they will love you.

I've had a couple successes hooking Twilio up to a CouchDB instance for SMS based data collection. This might be an interesting way to start using it.

Plus, if you aren't getting too sophisticated, you can just use Couch to handle the API calls to Twilio directly.

Write an app that reminds people to take their medicine x times daily.