What does the name after the "val" refer to? Is it the Valinor who infused the bloodline or is the name of the family that was infused? I'm a little unclear on this.

In general it appears to be the founder of the family line.

If you know any of the val'Sosi, it may be possible to find out where the name came from.

So I'm thinking that this is the surname of the family, since I know Assante is a last name in the real world and I think many of the rest are as well.

Virdan and Emman were brother and sister (first names). I'm not sure about the others.

The story seems to be that that was the given (or possibly Clan) name of the original Vals (or clansomething which became the Vals). So Lord Virdan gave rise to the val'Virdan. I seem to recall being told the val'Sosi name being given to the players by PCI, and I know of no IC significance.

What the "real" story is, we shall see. . .

So to be clear would Ishi be the name of a single person or and entire family? That's part I'm unclear on (although it specifically seems to say in the texts that Virdan and Emman are first names, as they are twin brother and sister---at least that's what they tell us, who knowss the truth of things concerning something that long ago).

Unknown at this time. The lore suggests given names but are little more than myth currently.

We don't know the real answer.

Most of the History before the Coryani Empire is based off of myth, tradition and some archaeology. That Virdan and Emman were siblings is the Val Emman story that emerged in/post Sibling Rivalry. Is it real? Maybe. But it leaves out the val'Tang (now ul'Tang) and any other lineages that might have died out in the Blood War. In the Canceri book the origin of the Virdan is told differently yet again....In at least one Val clan the name is supposedly the last name (clan name) of their founder.

So short answer...lots of competing myths and stories circulating. Be interesting to know how the preferred story of the val'Ishi... Or how the val'Vasik settled on Vasik.