Technically the Battle of Bunker Hill could have been called the "Battle of Breed's Hill". This is where the Continental Army built their main fortifications, and so most of the fighting happened here.
Breed's Hill had only one thing to recommend it over Bunker Hill - it was closer to Boston, which made it easier to fire artillery at the city. Though of course one disadvantage of this situation was that it was also a lot easier for Boston to fire back. Nobody knows why William Prescott chose the spot over Bunker Hill - it's possible that, since he was working on fortifications in the dark, Prescott didn't realize how vulnerable the hill actually was until the sun rose. By which point it was a little too late to up and move. I imagine that was probably quite an awkward moment.