Proserpina is not a character in the play but rather the Greek goddess Persephone, who is referred to in Latin and pre-20th century English as "Proserpina" or "Proserpine". She is mentioned by Perdita in one passage:
I would I had some flowers o' the spring that might
Become your time of day; and yours, and yours,
That wear upon your virgin branches yet
Your maidenheads growing: O Proserpina,
For the flowers now, that frighted thou let'st fall
From Dis's waggon! daffodils,
That come before the swallow dares, and take
The winds of March with beauty; violets dim,
But sweeter than the lids of Juno's eyes ...
The major literary sources for the story of Persephone are the Homeric Hymn to Demeter and the Rape ("seizing" -- the sexual meaning is modern) of Proserpine by Claudianus which was probably Shakespeare's source.
Persephone. while picking flowers, was seized by Hades (also called Pluto or Dis) and dragged down to the underworld. Each year as a result of a bargain between Demerter, goddess of the harvest, and Zeus, she spends half the year above ground (summer, when the crops grow) and half below (when Demeter morns and nothing grows).