From the name of the poster, I would guess that the name being referred to is the one used in North India.
I should add that to my list of Indian names whose meanings I am searching for. So far, I have Rita/Riya/Reena and now Rini/Rinku. I believe all these names are inspired by, if not copied from, non Indian sources in recent times.
Rini may also have been influenced by its onomatopaeic use as the soft sound of anklets of a girl dancing. Rinku may be a folk diminuitive of that.
All guesses, but who knows? (Why do all these start with Ri-?)