Are you talking about doing the job yourself or having a professional gunsmith do it?
Either way in order to remove slight pitting, the best method is to bead or sand blast it with a blaster. Then depending on if you want a matte finish or a polished blued finish you could either leave it rough and blue it or polish it down to a smoth finish and hot blue it.
Unless it's an old traditional looking sporter, personally if I were doing this with a gun I owned I'd have all the steel and other metal parts sand blasted to a satin matte surface then have it manganese parkerized on the steel to a very dark (almost black) charcoal gray, and then any aluminum alloy parts annodized black, if any. I find polished hot blueing to be great on older traditional guns or any gun if you live in a very dry climate, but I find phosphate parkerizing to be a much better and a more utilitarian finish for a pretty inexpensive price. If you want even better corrosion resistance you can have a baked-on or heat cured epoxy matte paint applied on top of the phosphate parkerized finish. The British military has utilized this method for decades on it's arsenal rebuilt guns and have had great results with it's durability and corrosion resistance.
Personally I've always though that traditional blueing is a pretty useless and worthless finish on gun steel, just my opinion. Though I admit that there are a great many blued guns still around with their original blued finish that are still in relatively good condition that have survived use in the old west and even the trenches of WWI just fine.
Historically, hunting has been a sport that has been predominately participated in by men. There have been notable exceptions, of course. Eleanor O’Connor, wife of the famous hunter and outdoor writer, Jack, traveled with him and hunted in many parts of the world, taking her share of game, including some exceptional trophies. Not as well-known to hunters today were Martin and Osa Johnson of the early to mid-1900’s. Together they traveled to many places that seemed extremely exotic and...