This is what Ancestry.Com has.
CONNOLLY Name Meaning and History
Irish: Anglicized form of Gaelic Ó Conghaile ‘descendant of Conghal’, a name meaning ‘hound valiant’ or of Ó Conghalaigh ‘descendant of Conghalach’, a derivative of Conghal; the two surnames have long been confused. Another possible origin is the West Cork name Mac Coingheallaigh (or Ó Coingheallaigh) ‘son (or ‘descendant’) of Coingheallach’, a personal name meaning ‘faithful to pledges’.
Dictionary of American Family Names, Oxford University Press, ISBN 0-19-508137-4
An error can be in your assumption if you feel there is only one Connolly family. When surnames were taken or assigned in Europe during the last millennium it wasn't done to identify a man as a member of a family but for taxation purposes. When they got through it wasn't impossible for legitimate sons of the same man to have a different surname and still each could have shared their surname with others with no known relationship. Most Irish and Scottish names are patronymics. The prefixes Mc, Mac, Fitz means son of. The prefix O means descendant or grandson of. Many have dropped the prefixes. But here is the situation, I had a person once asked me about someone named O'Brien if he was a descendant of Brian Boru. I simply stated, "I don't know." A priest named O'Connell said there was more than one Brian that had descendants. I knew someone in the line of O'Connor and she told me there was more than one Connor that had descendants.
A lot of people get taken in by surname product peddlers. There has recently been an ad on TV for a company selling framed surname histories which is somewhat shady for the reasons I stated above. The funny thing the man in the ad states "a" coat of arms will be on it, not "your" coat of arms. You see on TV the FCC can slap a company hard for fraudulent advertising. The FCC has no control over the internet or some merchant with a booth in your local shopping malls selling coats of arms(they use the misnomer "family crest" which is only part of a coat of arms). The surname product business if a big scam.
Now, we are no doubt all related if we go back far enough but as someone said your chances of being related to someone with an entirely different surname are frequently just as good if not better than your chances of being related to someone with the same surname. You have 2 parents, 4 grandparents, 8 great grandparents, 16 great great grandparents and it keeps doubling and eventually pyramids big time. There is a time the first part of the last millennium, the number of greats you put in front of grandparents, if they were all a different person, they would outnumber the population of the world at that time. So if you are married, you are no doubt married to cousin, it is just a matter of how many times your spouse is a cousin and if you can find the documentation to prove it.
If your question attracts someone to give you a link to a surname product peddler like House of Names or AllFamilyCrest realize they are just trying to sell you something. They sell coats of arms like they belong to everyone with the same surname and they don't.