I have to correct two things. First, Mark, the sequence is 3,4,2,1 and second, I just read that high pressure start air really comes at values of 2.700 and more PSI.
So disregard my statements towards such high values being wrong. Maybe I'll edit my posts then.  :-[ (done)


I think you go for the inboards (3 and 2) first because of the shorter supply ways of the external air connected (which often has a central point of connecting to the plane) or even the high pressure bottle.
External stuff usually has one contact point and then gets distributed inside the planes structure so the delivery ways differ in length. Length causes a loss of air pressure, so the shortest way has to be preferred.

If they are running then, pressure can be supplied more stable and reliable to the outer engines being self-sustaining then.

Outboard first might work too but as you have to start the inboards anyway, use them first as their supply will usually get the required "N"-values faster, more stable and with the mentioned shorter air supply ways.

So whatever happens, start the inboard ones first, must not be the No.3 then. The side might depend on the placement of ground vehicles, carts and personnel.
Our 707 gets started on the ramp, so obstacles and people are quite close to it.
"Did anybody see the guy with the Chocks?" "Yeah, he's stuck in engine No. 3!".  ;D

If it was a special need to go for the right inboard on the 707 first (meaning No.3 then), this might be explained with the external air cart to connect nearer to the No.3 than to the No.2 engine. Same "shorter ways" theory to apply then.

I don't see any special equipment reason to start 3 before 2 as they are connected to similar systems.
Other planes might need a yellow hydraulic circuit first or such things and therefor prefer special engine numbers to be started before the others. The 707 has some special equipment (the Air Compressors) but this is spread across No. 2,3 and 4.

Maybe Lou can help here.