If I remember correctly, there is a phrase meaning "a character suddenly gets a lot of development right before they die". However I have forgotten the name and can't find it. Perhaps I am just imagining that it has a name. Is there a phrase meaning this specific thing?
The trope you are thinking of is called Death in the Limelight.
Briefly, an episode or issue that suddenly focuses on a character specifically because they're going to die at the end (or fairly close to the end).
Usually this is a relatively minor recurring character, or someone who technically is in the main cast but never had a Backstory or much in the way of characterization. The episode/issue may be honest about what's going to happen, or try to make it appear that the character is finally going to be promoted to a main character. Episodes about the death of a major character who already got plenty of exposure and a fairly full Backstory don't quite fit this trope, since it's not as obviously different from the character's normal treatment.
Some other writing tropes that come to mind:
Developing Doomed Characters - Before the movie/book gets to the good parts, they have to spend time developing characters that nobody cares about and are going to die anyway. Think of the recent Godzilla remake, where a huge amount of time was spent on the human protagonist's backstory when everybody was just there to see monsters fight.
Death By Flashback - Right before a character dies, the story flashes back in an extended backstory sequence, develops the character, and ties up all the loose ends in their backstory, trying to make them way more interesting and their life more meaningful before they officially die.