You have the right instinct, narrative takes a backseat to player comfort. Don't worry if it makes sense in-game, that's not a real concern. Do whatever tying up of narrative threads you want, after you're sure the player is comfortable out-of-game.

If the player wants, just let them start the next game with a new sheet. If they want to do a sendoff, do it at the top of the session. If they don't want to acknowledge the old character, then they just evaporate and a new one is inserted.

Ultimately you're playing with friends, and the reason everyone is there is to have fun. The group's fun should take precedent over everything else.

That early on, you can just roll a hard mulligan.

Do a full character retcon, introduce the new character as if it were session 1, and maybe touch on how this character would have behaved over the last few sessions, then bam. Just keep running as normal.

This doesn't have to be complicated.

The old character gets a personal lead in another city, and they're going to follow it. They don't even have to tell anyone. They just leave.

New character conveniently enters the foreground shortly after.

I think you've received lots of great advice here already. One additional thing you may want to keep in mind as you're weighing your options is how the manner in which this character exits may impact other players.

Obviously, the needs of this player come first, and making whatever changes they need to ensure they can keep having fun in the game is most important. But I bring it up because, in a similar situation where I was running a game for a player who wanted to switch characters, I executed the change in a bit of a clumsy way that one of my other players found frustrating.

I had worked together with the player who wanted to change to craft a heroic last stand scenario in which their old character would sacrifice himself to allow the rest of the party to escape. Critically, we didn't discuss these intentions with anyone else at the table. One of my other players picked up on what was happening the moment we started playing it out, and did everything he could to prevent the loss of a party member. I essentially railroaded right over anything they tried.

Afterwards, they told me that while they understood why I ran it that way once the other player had a chance to explain their intentions and new character, it still wasn't very much fun to play through in the moment.

All this to say, if I had to do it again, I would make sure I have some sort of buy-in from the other players. They wouldn't need to know all the details, certainly, but I think a little heads-up would have made that experience much smoother for everyone.

You can jump ahead a few weeks in the future where the new character is introduced and deal with the departure in a series of flashbacks as you create tension to introduce your BBEG.

Few options available here I'd say, though I'm not aware of Vampire the Masquerade! Do the other players know that this player wants to reroll?

Definitely work closely with the player throughout the process and have them make the major decisions needed, though it's an excellent opportunity to drop a shocking moment on the rest of the party if they are happy for the character to die in order to feed the tension and danger of the story. It could be a good opportunity to introduce a huge villain in the story, a taste of what they'll eventually have to face down the line.

Other options could include taking something from the characters backstory that prompts them to leave the group in search of their own answers- retiring the character but leaving the option open for the future. Your players new character could be introduced as a prisoner or hostage, or even as an individual on their own mission that aligns with the current group.

There are so many different ways to approach, but the most important thing is ensuring the player is happy with the decisions. Having a little brainstorming session with them in private and putting a plan together you're both happy with is absolutely key though!

For my table, the goal is always maximum fun, so yeah if my player isn't enjoying the character they made - and you've already cleared the air with your player that it's not due to something else - then they should feel free to make a new one.

And there are so many cool ways to write the old character out of the story, especially Vampire. He joined the Malkavian branch of Scientology! Or maybe the werewolves found his safe house

I recently had two of my six players decide to change characters in between our third and fourth sessions of our Pathfinder game. In their case, it was because their precious classes weren't matching up with their expectations.

I told them it was fine and we could just handwave it and pretend their new characters were there the whole time. But one of the players took it upon themselves to write a really basic—but also very believable—excuse for the old character leaving and the new one joining.

We're all in this to have fun, so as long as they aren't constantly shifting around to the point it becomes distracting or difficult for other players' characters to form meaningful relationships, I'm more than happy to let a player change up their character if they aren't enjoying it any more. Session 3 is early enough that I'm happy just handwaving it, but if a player wants to write an explanation that's always better.