What happens if you attempt to change the name to a name that is already taken.

For example, Y'shtola Rhul is a restricted name which cannot be used, attempt to use it and see if it gives you an error (Name Taken Error) or the Cancel prompt?

Also, instead of clicking the Check mark, press the Enter key after selecting a name that is not taken.

Next, I would select a different starting class, such as Conjurer (different starting City), it should allow you to save and continue after setting a name, press Enter.
If nothing helps, try the process using a Gamepad (skips the Check mark button and uses Confirm by default).
I've had these issues myself, sometimes the Check (Tick) button does nothing on PC.