If bloodlines don’t matter, why would she keep a name that she has never gone by in the past and carries exclusively negative connotations for her? Why would Rey care if the “Palpatine name” is redeemed into something hopeful? What about that particular word is worth saving?
I think Rey keeping the Palpatine name is the worst possible outcome, and needlessly confusing to the audience from a thematic point of view. A good chunk of the audience has equated “Palpatine” to “evil” for longer than Rey has even been alive, and for her to suddenly accept the name after the whole climax revolved around her rejecting Sheev makes it appear as if she suddenly has some sort of reverence for him, which she absolutely does not.
Rey’s rejection of the Palpatine name is “because bloodlines don’t matter,” a symbolic rejection of the destiny of her blood and her personal choice to embrace another path. If she’s “Just Rey” (as some fans are trying to edit), then she’s forging her own destiny, and if she’s “Rey Skywalker,” then she’s taking up a name that does deserve to be kept in the light (and with Luke/Leia’s postmortem approval, no less). Either option makes more sense both thematically and in-character than “Rey Palpatine.”
And before anyone points out that Luke kept the Skywalker name: “Skywalker” was originally the name of a Jedi whose goodness Luke still sensed buried within Vader and which Luke managed to bring out before Vader’s death, not to mention Luke had already been named “Skywalker” his entire life. Rey taking the name “Palpatine” would be equivalent to Luke changing his last name to “Vader,” because evil is all that name has ever meant to them.