Which Side of the Lane Line?
 

Free throw is good to tie the game with five seconds to go. Time out request granted.
I, as the new trail, grab the ball and stand off the far lane line in OOB area to await the ensuing throw-in administration. No particular reason I went there instead of the near side. I just went to that side.
Teams come out and the thrower-inner says to me, "My coach drew up a play that has me doing a throw-in from the other side over there. Can we do this over there instead?"
What do I say?
A) No, I've been standing here the entire timeout. You're doing this throw-in from here.
...or...
B) It really doesn't matter, you can run the endline anyway. So here's where it's gonna be starting from right here.
...or...
C) Sure, it doesn't matter to me. Let's go over to the other side right now.
...or...
D) Other???
Anything by rule or by officials manual citation I should know?

I always allow them to start where they wish. In 2-man I would have no problem bouncing the ball across the paint. In 3-man, I will move to the side from which they wish to start the play.

No.

the general consensus is to move.

Interesting scenario. While attending a college officials camp recently, this event came up. The observer discussed with the officials, "what are you officials talking about during the dead ball?" The point was raised about where do you want the ball inbounding? The observer shared he preferred to take the ball out table side; to allow the other official across the court the ability to have both benches in their line of site, in case a coach wants to call a time out. He commented the official inbounding the ball is focused on the thrower and not the benches. Based on your OP: answer (D)

NFHS Rule 2.3 may come into play on this.

I am putting ball in table side. If he wants to move to the other side, he can once I give him the ball.

As am I. It did get me thinking...in two man are you going opposite since the inbounding team is on the other end of the court? That would give the T a better chance to see a time out request in their peripheral and the L a better chance of seeing/hearing a request since it puts him/her nearer the bench.

I cant help you with two man. Havent done or thought about two man mechanics in over a decade.

Very jealous of 3 man areas coming from the land of 2 man. BillyMac, your "Welcome to Connecticut" picture is requested. (my idea of humor)

Since I'm already tableside to administer the completed free throw(s), that's where I administer the throw-in.

I've never had a player ask to move to the other side, but he's entitled, under the rules. Since we're not allowed here to bounce the ball across the basket-line, I'd just move to the other side, and let my partner(s) know.

As the old lead/new trail coming out of a timeout after a goal, I've always just administered the throw-in on the same side of the court where I was when the timeout was granted.

The only exception is in a situation where I've wanted to get one official away from tableside, usually when I'm hoping to move myself/my partner away from a confrontational coach for awhile.

For example:
Coach gets whacked by my partner, who then moves to the opposite side of the court to administer the throw-in after the foul shots for the T. Next trip down the court, partner calls a foul during a try, reports, and stays tableside. Coach is still steaming, but not enough for a second T. If the timeout follows these shots, I'm going to try to set up so that my partner starts out opposite the table when we resume play.

I can only remember explicitly doing this once, but at the time, it seemed like the best course of action. By the time my partner got back tableside again, the coach had calmed down and the game finished without further incident.

table side
 

I can only speak to 3-person mechanics, but you want the center to have last shot responsibilities, as he/she typically has the best angle to see the ball leave the hand and the clock/light at the same time. In NCAA-W (center) and NFHS (opposite the table), you want that center opposite, so I'd recommend the trail staying table side.

Bye Bye Birdie ...
 

No way. Didn't you see what happened to mutantducky?

My opinion regarding this situation is this: Where would the Game Officials be if no Team Time-out had occurred? That is the Game Officials should be where the Ball is placed at the disposal of the Thrower.

MTD, Sr.

I will move, every time.

That's pretty funny.