Marilyn wants her family to remember her in a positive light, as someone who has always loved them. She doesn't want her family to remember her with regret.
In the story, Marilyn is a stowaway who will soon be released into space because the EDS (Emergency Dispatch Ship) she is on doesn't have enough fuel to support the extra weight on board. Marilyn originally boarded the ship in order to be able to see her brother, Gerry, who is stationed on the frontier planet of Woden.
Before her release into space (which will kill her), Marilyn engages in conversation with the EDS pilot, Barton. She tells him she wants her family to remember her as a loving person, one who performed innumerable acts of love for them in her lifetime, just as they have for her. She also wants her family to remember her as she has always been: young, whole, and unmarred by death.
She imagines that she will die a gruesome death in space, and she doesn't want her family to think of her as having died in agony. For his part, Barton assures Marilyn that her family would never think of her in such a way. With great kindness and sensitivity, he assures her that she has nothing to worry about on this score, saying,
You’re their own, their child and their sister. They could never think of you other than the way you would want them to, the way you looked the last time they saw you.