Maripryssmith: Surely the 'welshified' spelling should be Rhymni as the initial letter is Rh. Of course, this would seldom be heard in Welsh, even amongst fluent speakers, because the name is often subjected to the soft mutation, which drops the h, making your version right, but only in those contexts. Using a Welsh name in an English language song it is difficult to justify the mutation.

Hwyl dda (Cheers) Nigel