Under $350. Easy. Mossberg 500 Flex ($250 around here on clearance - couple 'adopted' boys have and love them), Any other Mossberg 500 (I love mine and one son and a DIL does too), Rem 870 ( I don't care for them, but my boys - 3 of 5 - love theirs), or a used Mossberg 835 - Oldest (I have 5 boys and 2 DILs) love his that I bought him.
Me personally, I like my Mossberg 500, and bought a Weatherby PA-08 that I bought for duck hunting. Buddy bought the same gun for ducks as well.
I have 5 boys, 2 DILs and a lot of my sons' friends, plus buddies. Everyone has been very happy any of the Mossbergs and 870s, plus the Weatherby.
Last thing I will suggest is, get one you won't feel bad about taking under water when duck hunting! In the last 10 days, have had two of the kids take two in the drink!
Last sage piece of advice - get Eezox - and use it on you duck guns! It WORKS!