Reduced recoil ammo is made for 7mm-08, .308 Win, .270 Win, and 30-06.
If she's stuck on .30 caliber, the .308 Win, using reduced-recoil factory rounds would be perfect for her. The short-action bolt guns are great for women, though watch out for synthetic stocks that can't easily be shortened to fit smaller statures. I'm impressed with the Remington Model 7 with a youth stock.
I looked for 7mm-08 reduced recoil rounds and can't find them in most places. As good a cartridge as it is, many of the big-box stores don't carry even the regular ammo in that cartridge.
Another round that may be under-rated for deer is the .243 Win. It has the advantage of flat trajectory that minimizes holdover at reasonable ranges, and with 100 grain or other "deer-sized game" rounds, is a quick killer.
The Savage 340 is an "okay" gun, but it a bit clunky with a scope, needing a side mount, due to the cut in the receiver top. They may be fine with receiver sights, but I haven't seen many on them. I don't recommend lever-actions for beginners due to the loading/unloading situation with gloves on and in the relative darkness/cold. I've witnessed several accidental discharges.