It's a special case of a more general (Israeli) bullshit detection technique. They'll ask something unexpected, but reasonable enough that an honest person would be able to say the answer immediately. The idea is that you bait a person presenting false information into revealing that they are lying. Security will ask a sequence of inane questions that a truthful person should be able to answer immediately. If a person being questioned grows shifty, nervous-looking, starts sweating, stops making eye-contact, or shows any other sign of illegitimacy, then the questioning will continue until (il)legitimacy is established. It is not a flawless method (see end of comment) but it usually works. It is virtually impossible to nail every single aspect of a BS story. If you're 100% legitimate, then you'll probably have no problems recalling things like your grandfather's name or the synagogue you were Bar-Mitzvah'd at. I'm a pale South African Jew - I was asked that.

Example:

Security: Are you a member of a Hebrew congregation?

Person: Yes, such and such a synagogue

Security: Who is the Rabbi there?

A legit person might say "Oh it's Rabbi so-and-so", but if a person hesitates, and has to think, and blurts out an answer like "Rabbi Cohen" then it is less likely they are being forthcoming.

In some sense, this is a very rough mental-Bayesian process of probability updating. A person arrives at the border / El-Al check-in. Security assigns some prior probability p_0 that a person should be allowed access, and via a string of questions, amongst other techniques, a posterior probability p_n that that person should be on that flight / in Israel is established.

As reitanqild mentioned, the LI is probably to help with basic vetting.

Family names - my guess is to perform a more rigorous background check. This is probably more relevant for people of Palestinian descent.

LinkedIn URL most likely was instead of a written CV. Wish companies would accept this.

As for the rest, I don't know but I hear they have some issues both with rockets as well as (often) clueless western demonstrants so getting being careful and getting good background information on visitors does sound like the right thing to do.