Here is some of what I have found. Some are iPhone and iPad apps; others are website-based. (I looked at more apps than I included here, but some of them weren't all that interesting or seemed to make sounds rather than music or musical tones.)
Here's the iPhone version. Dropophone (ADDED 6/16/10)
None of these generate music, just lyrics. But I anticipate that before long lyric generators and music generators will be combined so that you'll get finished songs based on the genres and subjects you select. Will any of them be great songs? Well, think of them like digital photos. You may need to produce a lot to get the right one, but you just discard those that don't work. And if you have something that almost works, you tweak it with the equivalent of a musical Photoshop.
And consider these for lyrics, too (there are actually far too many poetry generators in Google to list them all, but it should give you an idea of where to look if you're stuck when writing lyrics for a song):
Once you have found some lyrics you like, this application will create a song around them.
Songmaker for iPhone: "... simply speak the lyrics you want into the microphone while pressing the keys to enter the melody you want. After recording completed, SongMaker will play the song with your voice following the given melody along with background music." (ADDED 10/21/10)
I'm sure I have missed some good applications. So let me know about them. Again, I'm not looking for apps that require much training. This is a list for products which will allow the average not-very-musical person to create something worth listening to and perhaps sharing. I anticipate that as more products and tools are developed, the creations will become more sophisticated.
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This doesn't have to do with music, but if you are going start promoting your app generated music, you might need these to help you create your web page. 55 Astonishing Online Generators for Web Designers
Here's a video where two developers demonstrate a new, easy-to-use iPad music creation tool. They also explain their motivation. And they mention finger painting.
CultureLab: The first ever iPad music performance