Depends on the reason for acquiring a piece and how it might be put to use.
A build project rifle might give priority to action type since weight, stock, barrel length are all variables over which some control can be exercised during the build.
With a hunting blind rifle where shots might be made in more than one direction, a short barrel length might be the first concern.
A varmint rifle for prairie dog shooting might put priority on a relatively long and heavy weight barrel.
Sometimes price and availability trump the other considerations....that was the case with my first center fire rifle: a "sporterized" 1891 Argentine Mauser. And also with my last two acquisitions: a .460 S&W TC Encore barrel( MGM barrels were not available last fall) and a Ruger American 6.5 Creedmoor (sale priced comparably with a new TC Encore stainless barrel!).
Looking back, action type has typically been my first consideration followed by price. Bolt, falling block, kipplauf. Then size of gun kitty when something was available. The other five aspects are generally a wash. I did pick up a short barreled synthetic stocked bolt carbine based solely on weight once when afflicted with tennis elbow and handling an 8+lb piece was painful.
Rifle attribute priorities have also varied over time for me as I learn more about how these attributes impact accuracy, shootability, portability, and evolution of general rifle loonieness.
