Sounds like a short form like Rosmi, which is short for Rosmitah or Rosminah (at least 1 person uses the name Rosme as a short form for Rosmitah). Also found the name Rosmel (no meaning found).
In Arabian Rosminah رسمناه means "We have set" (online translation, so maybe not reliable)
I found the name Rosmita in be in use in Indonesia and
India. Perhaps Rosmita has a connection with the name Mita, meaning: "Fixed, Grounded". The origin of the name Mita is Indian (Sanskrit) (
http://www.babynames.com/name/MITA)
According to other sources Mita means "Friend" (
http://babynamesworld.parentsconnect.com/meaning_of_Mita.html)
There is also a name Mita in Italian, meaning "Myth" (from Italian mito).