There is nothing special about synthetic motor oil, use it as you would any other oil.
I've begun using 0w20 Mobile 1 synthetic as a gun lube. It seems to resist thickening up in cold weather better than other oils. Oil may not be overly important to use in a bolt action rifle but it can be important with semiauto shotgun as some do not like to be run dry. The 0 weight rating keeps it viscosity well below zero so far. As I tend to have a quart or two in the truck emergency box, it is always around if I need it.
I have a fair bit of "gun specific" oil around but the light weight synthetic motor oil seems to be at least as good as well as being much cheaper.
+1
I just top off the bottle from leftovers...If you change your own oil.......