I don't know anything about fishing in Italy so it's hard to recommend a fly selection. I like the fly box idea. We all need more storage for flies. I'm also guessing that the Brown Trout are of normal size (6 to 20"), so I think a nice pair of forceps and barb crimpers would be a great gift. Look at the Dr. Slick web site for ideas.
Does your son-in-law tie his own flies? If so, then a good pair of scissors and/or a bobbin make sense. I'm not exclusively promoting Dr. Slick products, but they do make good fly tying scissors and bobbins too.
Since you have to ship the stuff to Italy, these items are all fairly small and easy to package up. I imagine since he already fishes there that he has rod/reel/lines in good order. Plus shipping a rod would be expensive I think. If you knew what weight rod he fishes with, a nice Weight Forward floating line like Rio's Avid Trout series is a good choice. The WF5F (weight forward 5 weight floating) is on sale at Amazon for $60. Good fly lines are in the $60 to $90+ range.
Oh, another idea is fluorocarbon tippet in 3x, 4x, 5x, and 6x diameters would always be welcomed in my house, plus the spools snap together and would be easy to ship.
Wow ... I think I'll just copy this post and give it to my wife for Christmas ideas ... LoL. Good luck and I'm sure whatever you come up with will be welcomed and appreciated, CopperMan.