From the FIDE handbook
C.04.2 General handling rules for Swiss Tournaments
Before the first round the players are ranked in order of, respectively
FIDE-title (GM-IM- WGM-FM-WIM-CM-WFM-WCM-no title)
alphabetically (unless it has been previously stated that this criterion has been replaced by another one)
Thus whilst FIDE don't insist that the order be alphabetical, to avoid using this the arbiter would have to come up with some other criteria.
Further, under A4 we have
The FIDE Swiss Rules pair the players in an objective and impartial way, and different arbiters or software programs following the pairing rules should arrive at identical pairings.
which precludes the use of randomness at any point.
Is there a deterministic criteria for selecting order that would not tend to alight upon certain players for the bye in odd numbered fields?
(Sorry Ian - you posted whilst I was typing).