+1 to ride it first thing - or later in the day, the first time we went there we were staying in Livigno about 30 minutes away, arrived mid-afternoon on a Saturday and as the weather was great went to tick the Stelvio off the list, we went up the Umbrail (comes up North to South and joins just short of the top of the Stelvio meeting the West ramp)
Barely saw another vehicle, and by the time we came down again on the West ramp it was almost devoid of traffic as everyone was leaving as we arrived.
In 2013 we went over it again to "tick off" the East Ramp, we had come in via the Gavia and stopped overnight in Bormio so hit the Stelvio at around 8:30 and got a good run.
Tell your route planning mate to buy a copy of "Motorcycle Journey's through the Alps and beyond", it has @ 80 Alpine Ride-Outs covering around 300 passes with a number of good routes in the Stelvio region.
I have posted a bit about my two visits here:
http://www.alpineroads.com/phpBB3/viewtopic.php?f=3&t=1274
http://www.alpineroads.com/phpBB3/viewtopic.php?f=3&t=4143
Lots more good info on the site ::008::