Mrs. Rossini knew Billys grandmother and she was ill and we thought maybe Tony and Angela would look after him permanently but thats not true. They only foster parented him for a year and didn't legally adopt him. They didn't show this on the show after the Washington DC Episode but Mrs. Rossini found someone through her grapevine that has chosen to adopt Billy which why he wasn't around after Tony broke up with Kathryn and fell in love finally after 8 years with Angela.
Frankly, I'm glad that this show didn't make Billy a permanent fixture. I think this was the first (and maybe only) show that didn't fall on the cliche of bringing in the "cute" new little kid to "revive" the show because the original kids were *GASP! OH NO!* growing up. It never ceases to annoy me, even now in the millenium when shows continue to fall back on this tired grasp at a ratings booster. It was bad enough that they brought Billy in temproraily. I mean, look at all the shows that did this: Facts of Life (when Mrs. Gehrig's sister came in and adopted that boy who became the "cute little boy that all the grown up girls of the show fawned over because he's so cute"), Cosby Show, Gimme a Break, Diff'rent Strokes (Drummond married again, and oh look! She comes with a five year old son...). I'm sure there are others, I just can't think of them right now.
Growing Pains also brought in Chrissy Family Ties had a baby brother (name forgotten) Full House had new baby twins born (at least it "made sense" in their plot since Jesse and Rebecca were newly married and would have kids - one of the BETTER times this was done and for that show it worked better because the WHOLE show was always about the cute kids).
So many of the shows did this when the original show wasn't even focused on "cute little kids" (they were more about the adult relationships or teens) and still they did it.
The cute kids thing was annoying and way over done (and NEVER worked) - BUT one thing that was particularly crazy about the trend was how many of them had babies one year and the kid was in kindergarten the next. OR shows where they have babies (i.e. Friends) and the baby NEVER impacts the characters at all (Rachel just hangs out with everyone exactly as she did before with a rare mention here and there of the baby - no need for a sitter?).
Billy was one of the WORST cute kid additions and thankfully he didn't last - and thankfully it at least "made sense" in the context of the show (they didn't have a baby who is in kindergarten next year like so many).