Google Tasks look great, but I don't want to have to use Google all the time, just to see and manage my tasks, and there doesn't appear to be a decent API for Tasks. So you can't sync them to Thunderbird or anything else.
KOrganizer looks nice, but I don't see any good sync options there, either.
So I stumbled across Remember the Milk, which is also a web-based task manager, but with lots of other support. There's an Android app, and there's also an iCal URL you can use to view tasks in other places. The only downside is that the iCal support is read-only.
But I found a way around that in Thunderbird. First, you just add the iCal service as a calendar. (Get that URL from Settings> Info, in Remember the Milk.) That'll at least let you see your tasks in Thunderbird.
Now, how to add tasks? Answer: Use RTM's google gadget in a tab in Thunderbird. First, you'll need to download the WAT extension for Thunderbird. Among other things, this will give you a box in the upper right corner where you can enter a URL, which will then be opened in a new tab. The URL to enter is: