There are actually several townlands in Mayo called Rathbaun (in Irish, an Ráth Bhán), which is a fairly common name, but the one which Roger mentioned is the only one in the parish of Backs. You can see more info about that Rathbaun at this link, which includes an interactive map, allowing you to look around the area: https://www.townlands.ie/mayo/tirawley/ballynahaglish/mount-falcon/rathbaun/
As you can see at that link (through a further link there), the townland name was anglicized in various ways over the years, including as Rathbane and even Rabane (the "th" letter combination is now siient in Irish).
The Catholic parish records for Backs which are available online can be accessed at this link: https://registers.nli.ie/parishes/0738 , but those don't seem to go as far back as what Roger found. Perhaps he has another source.
The Barrett’s were an Anglo-Norman family which became fully Hibernicized (using the name Bairéid in Mayo). They became like a native Irish sept (or “sub-tribe”), having a well-defined territory on the eastern side of Lough Conn in northeastern Mayo, in the area of what are now the parishes of Backs and Attymass, and they intermarried with the other families in the area for centuries. One branch of the family changed their surname to Mac Andrew (Mac Aindriú in Irish) long ago, and I have McAndrew ancestors from the area myself. Later, the name was sometimes re-Anglicized as FitzAndrew. Not sure that will be of much help in your search, but if you turn up dna matches with the McAndrew or FitzAndrew surnames, you'll at least know the likely reason.