'Groove Thang': Meaning Thereof?

No idea. Can you give more context?

Sure,
They went to Ray's reunion and Debra had a good time dancing with other guys. Ray didn't like it and they started to argue as they got home. Debra mentioned about ray's Eustachian tub's problem and his sensitivity and Ray said: What is a big medical problem with you, ruptured booty? Deb answered: I never ruptured my booty, but one time I did sprain my groove thang.
I hope this helps, Jim.

Booty means her buttocks.

I should think that her 'groove thang' is her vagina.

A salutary warning to us all.

MrP

Nona has already answered.
I think I know why I never watch that show!

Thank you all.
I would have never thought in such a family sitcom they say such things.

Well, 'groove thang' would go far above the heads of any children watching.

I must admit I spent 10 minutes wondering what to put as the translation as I do not want to offend anyone, but decided it was best to play it straight rather than get into more confusion with further euphemisms.

I've never seen the show myself. You have to bear in mind though, what is unmentionable in one culture is quite ok to refer to, at least obliquely as in this example, in others. We are not very shy in England! Groove thang would just make me laugh and I don't consider it too strong for a family show. Just my opinion of course.

Is there a play on archaic dance-speak here too?

In years gone by, a dancer would sometimes 'get into a groove'. A 'groove-thang' might also be one's dancing groove. One might metaphorically 'sprain' it by exceeding oneself on the dance-floor.

Returning to Nona's diagnosis: Debra must have danced with exceptional vigour. What would you say to your doctor...

I say all this from a strictly sedentary position.

MrP