Voyager 1 was the first to reach Jupiter and the first to reach Saturn, as it was launched on a "shorter and faster trajectory" (Wikipedia, NASA). So the numbering was chosen to reflect the order of the main part of the mission, not the launches.
I have not found any sources explicitly stating that as the reason, but the arrivals at Jupiter and Saturn received much more publicity than the launches, and at the time it certainly seemed natural that the first Voyager mission to reach Jupiter would be Voyager 1.
In Exploring Space by William E. Burrows, there is the following footnote:
Three weeks before the scheduled launch, the spacecraft that was
originally to be Voyager 2 developed mechanical problems. As a result,
a "spare" that actually carried the designation Voyager 3 became
Voyager 2. Once repaired, the original Voyager 2 was made Voyager 1.
The original voyager 1 was shipped back to JPL.
This at least shows that the naming had nothing to do when when each piece of hardware became available, since they renamed the hardware when they had to swap it out.
A reference to a NASA memo or even contemporaneous media coverage specifically supporting this reason would be a better answer, but this is what I have.