Late night/early morning transport from Barcelona Airport to city center
 

Will be in Barcelona landing early morning 00:10 and need to get to city center hotel. There will be three of us with small luggage. Any recommendations for such a late time frame? What would a taxi cost? Anyone have a private driver suggestion?

A taxi would cost around €30, about a quarter of the price of a private transfer.

As Robert wrote. While midnight is not exactly a late time in Barcelona, I'd just take a taxi.
If you arrive at Terminal 1, there will be a central taxi pick-up area one level down from arrivals hall. The sign is huge so you cannot miss it.
At T2 you just go outside at find the taxi rank at the curb.