Hello
It is a very simple game, you have to choose a unique name where the first name should start with the initials of the last name and last name should start with the initials of the first name.
Let's say, I will give a unique name
Elina Abgail
Then your first name should start with letter A and second name should start with letter E. The one who will answer can put another name to carry on the chain.

(You misspelled First in the title)

Artemis Errant

I'm not sure if I understand, do I have to come up with another completely separate name for the next poster? Wouldn't it be simpler to just discard the given name and use the initial of the last name for the next given name and pair it with a brand new last name?

Elvis Fortunato

Yes we can think of that too.
Fyn Thomas

Terrence Barrens

Benward Aaron.
Actually he's my favorite. I saw his name on babynology.com too.

Adrian Schmidt

FISRT

Lsat?

Leshat Lestrange