I think in any other game, Rebirth would be a fun, cute card that would let you think your way out of particular match-ups, or transition between phases of the game.

For example, Rebirthing from Kabonesa Wu into Jesminder or Kit after getting set up is a neat way to spend your influence and deck slots. And the ability to choose a different identity based on the situation offers a lot of fun decision points.

However, I think this design is a total mistake in Netrunner, for a couple reasons:

1. Identities that are blank after the start of the game: Identities like Andy, Val, or even The Professor have text that is only active when the game begins (or before) and are essentially blank afterwards. Because these Identities exist, it is a no-brainer for them to run Rebirth (except maybe the Professor) because it is always useful to them, even on turn 1. They have nothing to lose and everything to gain from including it in their deck. Compare this to Kabonesa Wu, who would need to hold onto Rebirth in hand for at least a few turns before using.
2. Identities are balanced around link and influence: Some Identities have stronger abilities than others, and as such have been balanced in other ways, namely now much link and influence that identity has. Omar Keung has a very strong identity ability, and as such has only 12 influence. This is meant to curb decks that use that Identity, because if he had 15 influence, he would likely be considered stronger than several other Identities. Kit has only 10 influence, because her ability to turn ice into code gates is very strong. Rebirth lets you play as these notably stronger identities, who are balanced with a downside, without the downside.

Because of these two factors, Rebirth allows Val, who has a strong abiilty that is notably blank after the start of the game, to use Omar's powerful identity ability. The player receives the upside of Val's ability and influence, with the upside of Omar's central-targeting power.

This, combined with Rebirth's "Limit 1 per deck" clause results in swingy matches where sometimes you're playing against Val (who is just fine in her own right) and sometimes playing against Omar with 15 influence, and you have a bad pub. Sometimes you're playing against Andy, and sometimes you're playing against Leela who got to start the game with a 4 extra cards in hand. For these reasons, I think Rebirth is a good example of bad design. Unfortunately, being poorly designed and slightly frustrating is not enough to warrant Removal from the game for now.

As mentioned in valerian32's review, this card is likely to replace Scavenge if that card rotates along with Creation & Control (as is allegedly going to happen at the end of 2019).

Rejig is interesting because it allows you to achieve a couple of niche effects. Two that seem most useful are re-installing programs that come into play with counters (Pelangi, Imp, Cerberus "Lady" H1, D4v1d) as well as allowing Kabonesa Wu: Netspace Thrillseeker to keep cards she tutors in play. Unfortunately, Rejig has yet to see much use in serious decks. The modern praxis is that it is typically easier to sacrifice spent cards to Aesop's Pawnshop or even Spec Work, and then recur them as necessary with Clone Chip (or simply install a new copy). Wu typically uses Self-modifying Code or Scavenge to get around the downside of her identity ability. Clone Chip and Scavenge simply have greater versatility compared to Rejig, and as such see much more play, even in Anarch (and sometimes Criminal).

While Rejig hasn't been seen in top-tier play, beneath the surface, in the seedy Netrunner underbelly of the jank-pits, Rejig does enable one trick: re-hosting icebreakers.

In decks that attempt to build a mega-breaker with Dinosaurus or Baba Yaga, drawing your icebreakers before the cards they should be hosted on is not ideal. However, Rejig essentially functions as copies 4, 5, and 6 of Scavenge, and allows you to easily uninstall your breaker, then place it on your newly installed dinosaur friend or spooky AI. This allows these decks to be on track to steal their first agenda by turn 14.

In casual play, it allows you to move Savant from Dhegdheer to Dinosaurus, a move that instills fear in the heart of any Corp, granted they have not already scored 5 points by now.

TL;DR This card is worse than Scavenge, but is neutral, so there you go.