One of the original smoosh names, this amalgam of two classics projects a sweet, somewhat old-fashioned sensibility; it could come back as an aromatic nature/herb name.
In ancient legend, Rosemary was draped around Aphrodite when she rose from the sea. It was regarded then as beneficial in strengthening the memory and became a symbol for faithful lovers. In Hamlet, Ophelia says "There's Rosemary, that's for remembrance; pray, love, remember."
The name was long associated with George Clooney's aunt, the great pop singer Rosemary; less pleasant associations are with the 1967 movie Rosemary's Baby.
A common variation is Rosemarie; many parents are now looking to the stand-alone nickname Romy.