Is it possible to rename your characters after starting a game?

If you enable the in-game Lua console, you can use it to rename a character on the fly.

To enable the console, you need to edit your grimrock.cfg file, which can be found in the same place as your saved games. (You should close Grimrock before doing this.) The file is broken up into sections; in the first section, there's a line with console = false. Just change the false to true.

You probably also want to change the consoleKey line, since the default is a key that doesn't exist on an American keyboard (I believe it's the § key on a Finnish keyboard, but I'm not certain of that). I recommend using a function key, since whichever key you choose overrides anything else you might want to do in the console (for example, if you choose Z as the key, you cannot type a Z in the console). To use F12, change the consoleKey line to read consoleKey = 123.

Once you have the console enabled, you just have to type the following command in to change a character's name:

party:getChampion(2):setName("New Name")

Where you replace 2 with the champion to change (in the range 1 to 4) and New Name with whatever name you choose.

If you want to disable the console again afterwards (since it could be a temptation to cheat in all sorts of ways!), you can just change grimrock.cfg back to console = false.

According to discussion on Grimrock.net and a couple of other websites I found, it doesn't appear that there's yet any official way to rename characters after you start the game. There are quite a few people complaining about it as a missing feature.

However, one of the forum members created a python script that claims to do that. At the moment there is only a single reply (saying 'good job', so that's positive) but it was only posted little over a week ago. I have not tested, verified or otherwise examined this script so use it at your own risk.

And, of course, back up your save before you try anything!