Donal O'Sullivan's Carolan: The Life, Times and Music of an Irish Harper has this to say about 'the form of Carolan's name' (it's a long paragraph, so I've broken it up a little):

'His full name in Irish is Toirdhealbhach Ó Cearbhalláin, though one often meets with minor variations in the spelling of both forename and surname. The forename is generally rendered in English as Turlough or Terence. When Gaelic names of this type are written in full, it is proper to prefix the Ó to the surname. When, however, the surname is used singly, the only satisfactory method is to follow the form used by the owner of the name and by his friends.

'Carolan twice brings his own name into his poems – in his song for Fallon and his song for John Stafford (see the notes to nos. 46 and 161), and in both cases he employs the form Cearbhalláin. MacCabe, his most intimate friend, twice mentions his name in his elegy and uses the same form. Charles O'Connor, also a very close friend, refers to him in his diary, as Cearbhalláin and in his letters to Walker, written in English, as Carolan, not O'Carolan. It is therefore certain that Carolan was known to himself and his friends as Cearbhalláin, or, in English, Carolan.'

However, the village of Keadue in County Roscommon, where Carolan (following Sullivan) is buried has an O'Carolan Heritage Park, the O'Carolan B&B and, of course, continues to host the O'Carolan traditional music summer school and O'Carolan Harp Festival. So, in order not to confuse readers I have called him O'Carolan in the latest edition of the Rough Guide to Ireland though referred to him as Carolan in the Rough Guide to Irish Music.

Other well-known bearers of the surname include the current head of ITMA, Nicholas, who simply uses the Carolan form when providing his surname in English.

Sullivan does investigate whether or not Ó Cearbhalláin was a clan name, but does not come up with any explicit confirmation.