What does the O' mean in O'Brien?

Answer 1

That is called a patronymic, meaning a name was composed based on the father's name. Before families had names that would be passed on intact to the next generation, a person was sometimes known by his own name, a form of his father's name, and/or a form of his grandfather's name. So if three generations, from oldest to youngest, were Brien, David, and Fred, the youngest might be known as Fred O'Brien or Fred Davidson, or Fred Davis or Fred MacDavid or the whole Fred MacDavid O'Brien, depending on local custom and how many in the village he might be confused with.

When families did adopt family names to pass down intact, some of these same forms were applied to families. In other circumstances, other families might adopt the name of their occupation (Farmer or Carpenter), or the place they lived (Woods or Hill).

Answer 2

It means completely nothing.