A reason for this can be that the Widget configuration data (that is what is stored into the database) is mixed-up on that certain blog. Those cases are hard to debug. What you can try is to reset all widget settings on that blog. Sometimes it’s enough to remove the Text-Widget from the lower pane if it is available there. If not, remove the widget configuration from the database (backup first!). After that, the Text-Widget should be offered again.
Another problem could be something related to javascript. That’s just a guess, but to play safe here, just disable javascript in your browser and then use the widget screen. It has a new design as it offerers now widget configuration in a non-js variant. You probably can setup the TextWidget here. So try this first even if you do not have a javascript problem.