My thoughts on the meaning of kyr'tayl.
Consider these words ...
gaa'taylir is "to hold in the hand"
haa'taylir is "to hold in the eye"
kar'taylir is "to hold in the heart"
partaylir is ... not as obvious; perhaps "for preserving" or "victoriously preserve"?
With most of the words relating to "to hold in" something, the obvious meaning is "to hold in the end." Less obviously, it means "to preserve an end." Even less obviously, but certainly the way that one discussion I had went, it's an extreme contraction of a statement, as is the case with hukaat'kama.
One point to remember is that the phrase is ni kyr'tayl gai ... The person doing the adopting is not holding a soul or a person or "you"; they are holding a name.
This adoption vow, along with the daily remembrance of the dead, show how important names are to Mandos. Your name represents who you are. This is the Mando equivalent of the "true name" in functional magic.
In Triple Zero, Kal claims that Mando fathers name their sons. However, I think this refers to newborn sons. If I remember correctly, Kal received at least his last name from his adoptive father, but Walon Vau kept his full name when he was adopted. (In Hard Contact, Vau thinks to himself that he received a loving father too late to be meaningful; I'm assuming he's referring to his adoption as a Mando, but that's not directly stated.) Thus, I think there's a cultural idea of a "true name" of sorts being an important part of one's identity.
Another point is that the phrase ends sa (ner) ad, like my child, rather than bal cuyi ner ad (or some shorter equivalent of that), "and are my child." This to me says that there is at least a subtle difference between being born into a Mando family and being adopted into one. Perhaps it is simply a recognition that one's genetics are not being passed on, even if one's culture is.
But how does kyr fit into this? I agree with another language enthusiast who suggested that kyr in this context refers to becoming part of the Manda after death.
Combining the ideas of the "true name" and of kyr referring to the afterlife, ni kyr'tayl gai sa'ad could be understood to mean: "I give your name a place in the Manda as if I had given you the name as my own flesh and blood."
My proposal (with no knowledge of the original author actually intended): kyr'taylir means "to preserve a place in the Manda for (someone)."
Others thoughts on the meaning of this (though there is much more debate that I've heard about but don't have links to).