Hi Marilyn , yes you are correct , on the IGI O'Dwyer does not come up when you search Dwyer (almost evrything beginning with Dw does !! )
There are only 2 BMD O'Dwyers in Scotland and 81 in England . Many of course in Ireland. Marilyn , you have to be careful with the IGI for Ireland , virtually none from church records , most are from LDS members , so you have to be wary , I have seen so many incorrect postings by members.
Had a look for Michael Dwyer and found one Jan 26 , 1830 to John and Margaret Ryan in Tipperary. Looked for father John because of first born son of Michael - John. The Manx and Scots followed this pattern , and the Irish to a lesser degree. For your information , of my eight great grandparents , the only one that I can't find birth details for, is John Dwyer. He married 1865 in Scotland , his parents Andrew and Mary McManus , he had a brother Thomas. All censuses , marriage and death certificates just show "Ireland" !!!!!
Two of my cousins in Scotland have spent a lifetime trying to track him down , but nothing yet .