TBH, this could go in the English or Irish forum, but I chose to put it here, since this is a mistake that is almost exclusive to English speakers and Irish language names are inevitably buried in a sea of English.
Take a relatively famous Irish language name like Dara Ó Briain. For argument's sake, we can take it that the surname is historically anglicised as O'Brien (from the Irish, meaning 'Grandson of Brian'). But under no circumstances should anyone be writing O'Briain, not to mention the print overload of Ó'Briain.
Ó (sometimes archaic spelling Ua) comes from an old Irish term meaning 'grandson' and has no relationship to the English abbreviation o'/of (Will o' the Wisp, Sheaf o' Wheat, Cat o' Nine Tails etc.), even though English civil servants seemed to have come to the conclusion that it was, in their dealings with Gaelic-Irish names over many centuries. So you wouldn't believe how ridiculous it seems to someone even vaguely in the know, when people start applying English punctuation on an Irish language name.
It's actually a case of mixing the punctuation/grammar of two completely different languages together in a big mash and before anyone states that it somehow doesn't matter, you can be damned sure you wouldn't do it in the spelling of any other language (How about a hyphen in 'Von-Hindenberg', just for the laugh). A lot of its usage seems to be based on nothing more than "sure it's only Irish, so who gives a sh*t."
So spare a though for that poor little Ó and try not to burden it with extra un-necessary shoulder baggage.