Browning firearms, particularly their .22s will always have value. How much depends on condition, model, age, and use. The .22 rimfire round is cheap to shoot, accurate out to about 150 yards and certainly great for informal target shooting, small game hunting and even defense in a pinch. The Brownings however are classics. They are a high end product and in good condition can command a premium price. I would not however view selling it as an option as your son or some day grandson will strongly regret its loss. This is one of those things that rightly should become a family heirloom.
If you want a .22 pistol, spend $200 or so on a Ruger .22 or Browning Buckmark and keep the rifle. If it is older it would be worth far more than that and its centimental value puts it even higher as a keeper.