Basic Trope: A death that defines the protagonist's origin won't be undone.
Straight: The death of Abigail's father, Brian, due to a Time Travel experiment gone wrong makes her dedicate her life to perfecting his technology to save him. Once she does so, she realizes that if she saves him neither of them would have the necessary desperate inspiration or drive to solve the problems and there would be no Stable Time Loop, so he has to remain dead.
Exaggerated: Abigail perfects time travel after her father's death to save him, and manages to save countless of secondary and incidental characters using Time Travel but because of a poorly contrived Timey-Wimey Ball Brian has to stay dead.
Downplayed: Abigail visits an Alternate Universe where Brian is still alive. She decides that knowing he's alive somewhere is enough and stops torturing herself.
Justified: The way Time Travel works makes it a practical impossibility to rescue Brian in any way.
Inverted: As soon as Abigail decides to dedicate her life to perfecting Time Travel to save Brian, it's already as good as happened and a future version of herself appears with a time-displaced version of Brian, who is instrumental in helping Abigal perfect Time Travel.
Subverted: Abigail manages to save Brian, and she goes on to become a more compelling character now that she has to deal with the emotional transition of having a now living father who is younger than she is living in the present. Brian, for his part, is a Fish out of Water who alternates gratitude and resentment at being saved against his will.
Double Subverted: Time Travel can't change the past, only create an Alternate Dimension. The original/adult Abigail who saved Brian never gets to live her childhood with him.
Parodied: There's a support group for characters whose dear ones have died but can't be resurrected, while everyone else can come back from the dead just fine.
Zig Zagged: Because of the finicky nature of time travel, sometimes Brian can or can't be saved.
Neither Brian nor anyone else important to Abigail dies to give her her motivation.
The life, death or resurrection of Brian has no effect on Abigail as a character.
Enforced: The author knows that allowing death to be trivially easy to undo cheapens the drama, so he applies You Can't Fight Fate to the setting. Time Travel / resurrection magic can't affect people who already have an important destiny nor those who put them on that road.
Lampshaded: "Wow, I spend my life looking for a way to go back in time and save my father, only to find one that works for everyone else but not him.
Invoked: Cathy, Abigail's mother, wants to keep her from opening an ethical can of worms by implying her messing with time like that might cause Clock Roaches to come and eat Brian.
Exploited: The Super Villain Century-Aeon goes around murdering scientists who experiment with time travel so that their children will try to save them, just so he can taunt them that it's impossible.
Defied: As a child, Abigail discovers the reason her dad is alive is because she came from the future to save him. The mere threat of losing him is enough to motivate her, and the story focuses on her trying to close the Stable Time Loop with the help of her still living dad.
Discussed: "It's like Death has a sick sense of humor, the more important someone is to you, the harder it is to bring them back."
Conversed: "Why would an author make undoing death so easy but not let the main character save those she loves most? That's stupid and cruel."
Implied:: Abigail suspects that the reason she and other self-made time travelers have this problem is that it stabilizes Time and reduces the odds of a Time Paradox.
Deconstructed: The cruel irony of a world with easy resurrections for everyone but those most likely to want them and best able to get them leads to Abigail doing a Face-Heel Turn and murdering people left and right in the cruel knowledge that none of it can be undone— anyone clever enough to make a time machine to try and stop her, by definition, can't.
Reconstructed: Nothing is set in stone, Abigail can Screw Destiny and save her dad... but doing so means she deprives the world of a working Time Machine, and after looking around she realizes there's no one else who could replicate the discovery. She decides that, painful as it is, she must sacrifice her dream in order to honor her father's memory. Afterwards, she gains a level of inner peace, and becomes a more Rounded Character.
Played For Laughs: Brian is Abigail's Living MacGuffin, and every episode she tries a new zany time travel scheme to save him, only for it to fail comically.
Played For Drama: Abigail's continued failures leads to her despairing, only for her to smash her time machine— which causes the accident that killed Brian in the first place.
Intended Audience Reaction: The author wants the audience to realize something screwy is going on, and launches into a plot arc about the behind-the-scenes manipulations of Century-Aeon to keep Abigail from saving Brian.