A few points to consider: If the rifle is for your son to plink with, is it really worth the cost of a professional re-blue, including any post and packing charges? For very little more than these costs you could get a perfectly respectable new rifle for him to use.
However, if you really want to renovate the Falcon then, if it was mine and I wanted to do it up as a plinker, not as a collectable in a collection, I would strip the stock back to bare wood, stain it with something like a Peruvian Mahogany stain, add Danish oil for maybe 6 coats and end up with a decent looking, glossy stock. If you do not want to use Danish oil... why not a good durable varnish for the plinker? The action I would cold blue (cost of cold blue preparation approx £7.00 and you can do more than 1 rifle with it) or, if desparatee, a spray on black coating ... but cold blue would be better and more satisfying to do. Factor in any spares you might need, but most you could do yourself e.g breech seal, piston washer. You may or may not need a new spring but I suspect the spring in the rifle would be OK for short range plinking.
The biggest cost of all this would be your time and I suspect you could do most of it in a day. The stock finishing would take longer as the stain/Danish oil/varnish needs to dry between coats.
Cheers, Phil