There is nothing in the Boeing 737 book that says which engine should be started first, except of course for one or two of the Supplementary Procedures.
It makes sense to start the engine on the same side as the ground engineer, away from the side of latest activity and which can provide conditioned air the soonest.
As a point of personal interest, I've noticed that that Engine 2 start EGTs are generally higher which I think is due to starting on APU bleed. Starting No. 1 first on hot turnarounds and/or where the aircraft has a bit of a tendency to hot start (nearly always No.1 again) and even a crossbleed start on one aircraft that was going through a time of being particularly prone seem to work well.
Last edited by Sciolistes; 7th Apr 2010 at 11:57.