The refrain of the song you posted is the children's song I was thinking of, but with different lyrics. I heard it on some record. It's not from my or my children's childhood. Unfortunately, I can't recall the words...
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This thread is a reminder to me not to be an absolutist about the orgin and meaning of a name. I should have remembered that there are often names [and other words] with same spelling or similar spelling in different languages which have completly different meanings. And nicknames often are given that appear to have little or nothing to do with the spelling or pronunciaton of a person's name.
Which leads me to the name Rhiannon. I googled that name and learned that Rhiannon, Celtic goddess of the moon and inspiration, the enchanting fairy princess who rode so swiftly that no horseman could catch her, loved and chose to marry a mortal king of Wales.
Accused of murdering their infant son, Rhiannon bore her humiliating punishment with a grace and dignity that melted the hearts of her adopted countrymen.
Eventually proven innocent, Rhiannon was reunited with her husband and son and restored to her throne.
Later the Celtic goddess Rhiannon became the famous Lady of the Lake who, in the legends of Camelot, gave Arthur the magical sword called Excalibur."
I also found the similarly spelled Arabic name "Rihana" {variants: Rhiana, Rhianna; Riana, Rianna} in a number of baby books on names. Rihana means "sweet basil"
Btw, I also know an African American pre-school girl whose nickname is "Nonnie". She has a Swahili name that begins with a "N", but in my opinion, that name does not at all sounds like that nickname.
Again, this thread is a reminder to myself that at least in most contemporary cultures, there are no hard and fast rules when it comes to giving people "first" names and nicknames.