I knew that Braid was a video game and that a shard of console had been found, that was my conection to the video game, But I don't know when they first called it operation braid....
The earliest reference to "Operation Braid" that I can link dates from Tuesday 21st December 2010:
http://swns.com/news/parents-of-missing-25-year-old-bristol-woman-launch-plea-for-help-12546/The code name is given in the last paragraph of the news report in this link. I used to have a reference from the day Joanna was reported missing itself, that was a Monday, but it seems to have been removed from the internet.
No explanation for this cryptic code name was made public. I cannot think of any reason why it should have been prompted by the discovery of part of a console in Joanna's flat, as there are presumably many video games and many other possible ways to assign a name. Of all the names in all the world, why did the police choose "Braid?"
The link to a description of the game that I posted earlier today, in the post that was removed entirely, is:
https://en.wikipedia.org/wiki/Braid_(video_game)
There are three significant points about this:
(1) The game is about the search for a princess who has been "snatched". If Joanna had left of her own accord, then "Braid" was a tasteless choice of name. Therefore it seems probable that the police were already convinced that Joanna had actually been "snatched" - as indeed we know she had. It seems probable, therefore, that Greg Reardon had told the police that he had found the evidence of a struggle in the flat, which he would later describe in court. However, in public, this was left open. Not until Thursday 23rd December 2010 did her parents make public their own belief that she had been abducted.
(2) The central person who has been snatched is a "princess". Joanna's disappearance rapidly became a topic of such national importance that it was as if she WAS a princess. The Policeman subsequently defended all the publicity given to her disappearance by the need to mobilise the public in order to report anything and everything that might help find her. It seems probable that this massive publicity plan was already at the front of their minds when they decided to use the code name "Operation Braid".
(3) A peculiarity of this game is that the character of each of the protagonists is volatile and can change radically in the course of the play. Those searching for Joanna and those searching for her killer certainly came and went like the pieces in a kaleidescope.