The Lynch surname is reasonably common in East Mayo, Sligo, and Roscommon, but it also exists in several other parts of Ireland. The name has two separate origins.
The largest group of Lynch's is believed to descend from the Norman de Lench family, which was one of the tribes of Galway, so at least some of the Lynch's in Connacht are likely to be part of that group. There was/is also a Gaelic surname, Ó Loingsigh, which was anglicized as Lynch, Linchy, and Lynsky (with varous spellings), and is based on the Irish word loingseach, or "Mariner". It originated in several different places, so there are unrelated Lynch families in the areas of Antrim/Down, Cavan, Clare, Cork, and Tipperary.
Now that I think about it, the Lynsky/Lynskey forms are especially common in Mayo and Sligo, so some of the de Lench family may have switched to using Ó Loingsigh long ago (or Lynch's from elsewhere may have come to Connacht ui the 1600's) . As people continue to reassume the Gaelic forms of their names, the Ó Loingsigh form seems to be common, though the people invlved may not really be certain about whether their ancestry was Gaelicc or Norman (actually, after so many centuries, it could easily be both).