So, which Drow House kidnapped Halsin?
Fuck it. This morning @adelar-plays posted this excerpt of the dialogue you can have with Halsin about his time as, well, a sex slave to the drow. tl;dr: If you play as a drow with a noble background in the game, you get some extra dialogue options and through it learn that the House Halsin had been captured by was not just destroyed, but is looked upon very poorly by other drow. Badly enough that working for any surviving members might get you into trouble.
And my very first thought about it was: Wait, was he captured by House Oblodra? (Aka: Araj's House?) Sure, Araj does not react to him or he to Araj (Or does he? Has anyone tried?), but Araj's dialogue does imply that she was fairly young when House Oblodra fell.
However, there are other options of course. Let me go through them all.
Now, what we know: Halsin is about 350 years old. This happened when he was a "young druid" (though your guess is as good as mine what he might consider "young"). And it definitely happened before the Shadowcurse was cast. So at the very least he escaped the Underdark more than a hundred years ago.
So, the drow house in question must have fallen between about 300 and 100 years ago. That much is fairly certain. So, between ca. 1190 and 1390DR.
Before I go into the list of possible Houses, let me quickly go into what the houses are:
The drow are a matriarchal people, and usually group themselves into Houses. The most apt comparison for those Houses - at least among the drow of Menzoberrazan - is basically mafia families, that are usually organized around one matriarch. Most of the Houses have their own little army, and just tons, and tons of slaves. (Look, in Menzo there is about 5 slaves for every drow living there. Most slaves are goblins or minotaurs, but there are also other races.) The Houses are in constant rivalry with each other to become the most influencial House of the city. And if you as a drow do not belong to a House, you are basically worthless.
So, which Houses were disbanded in the timeframe?
House Celofraie got destroyed in a conflict with House Agrach Dyrr in 1367 DR. We do not know the reason, but it was probably simply a rivalry between the Houses. The same goes for House Mlin'thobbyn and House Syr'thaerl.
House Do'Urden is one that we know a lot about, duh, because Drizzt is such an important character for the lore. The House was originally very important, but lost the favor of Lolth after Drizzt saved a normal elven kid on the surface. This led to the destruction of the House in 1338 DR.
House Elec'thil got destroyed at some point between 1327 and 1367 DR. The exact reason was never given.
House Bron'tej got destroyed in 1227 after a conflict with House Barrison Del'Armgo.
House Kenafin and House Horlbar were disbanded in 1383 DR, though the members of both Houses together formed the new House of Melarn.
House Srune'Lett nearly started a drow civil war and got subsequently destroyed in 1372 DR.
House Tuin'Tarl got destroyed in 1383 DR, after trying to attack House Kenafin and House Horlbar. In a revenge attack, obviously.
House Teken'duis had a rivalry with House Freth. When trying to assassinate their leaders, they failed - and in return the family fell into shame and was susequently executed by the ruling council of the city. This was in 1319DR.
House DeVir formed an alliance with some gnomes in an attempt to overthrow one of the other houses. Because Lolth hates everything not-drow, they lost the favor of Lolth. House Do'Urden took advantage of this and destroyed the House in 1297.
House Oblodra, as we all knew, allied with some illithids. Now mind you, this was seen as somewhat okay for about 2000 years, allowing some members of the house to develop psionic magic. Until some members of the members of the house got about the same idea that Gortash and Durge had: "Hey, if we use magic on the tadpoles and put them into people's brains, we can mind control them and take over the city." Which went about as well as it went for Gort and Durge, ending with the destruction of the House and Exile of all members in 1358DR. It should be noted that this is the one house, where Lolth took an active role in their distruction, sending demons to destroy them.
So, based on this information and the dialogue that was shared...
Again, I will disregard the entire "when I was young", because your guess is as good as mine to what Halsin considers "young". But the other information given there is more interesting: The fact that if you associated with the House, you might lose the favor of other drow/might actually get into trouble.
Because to me that tells me that the House in question was not destroyed in a rivalry. So we can disregard pretty much all of the Houses that were destroyed in a simple conflict with another House of the city without any further information given. It makes me assume that the House in question has lost the favor of Lolth, making them outcasts among the other Lolth-alligned drow.
That leaves the potential candidates of:
House Do'Urden
House DeVir
House Oblodra
With the vague possibility that it could also be Teken'duis. With them it is not quite clear whether they still have the favor of Lolth or not.
So, yeah. It could obviously also have been a not before named House in the city. But if it is among one of the Houses in question, it should be one of those three (or four).
Make of that what you want. One way or another, there is a chance that either Drizzt, Viconia, or Araj might have met him - even if they do not remember him necessarily. Given that to them he might have just been "another slave".
Bonus:
Since we know that he was with the House for 3 years, before he escaped in the chaos of the House being destroyed, we can also calculate Halsin's age at the time.
If he had been a prisoner of House Do'Urden, he would have been around 190 to 200 years at the time. Had he been a prisoner of House DeVir, he would have been between 150 and 160 years. And had he been a prisoner of House Oblodra, he would have been 210 to 220 years old. (Just rough estimates, as I do not assume he is exactly 350 years old.)
