Is this a sealed cartridge bottom bracket? If it were a serviceable cup-and-spindle bottom bracket, you could take it apart yourself, repack it, and adjust the bearings properly. You can't do much with a cartridge. How do the cranks attach to this bb? Play could either be in the bb itself, the bb attachment to the frame, or in the crank-spindle attachment. I would assume neither of the latter is true if the mechanic checked it. Personally, I wouldn't keep riding a bike with play in the bb. The front derailleur won't shift properly and it will wear your chain quickly and feel bad. What brand of crankset do you have, and is the bb a traditional square-tapered spindle, ISIS, or something else?