Rey shouldn’t have taken the Skywalker name at the end of the movie, but not for the same reasons you might think. Yes, Rey made out like a bandit – she got Luke’s family farm, Solo’s Falcon, and Poe’s droid – but the one thing she could have taken, didn’t, but should have is the name Palpatine.
Rey taking the Skywalker name fits with the themes in the new trilogy, but not as well as Palpatine’s. Taking the name Skywalker is still Rey running from her past and denying who she is and what’s inside her. Taking (or keeping) the name Palpatine, Rey not only accepts who she is, but she also chooses to be a Jedi in spite of her lineage. And, the ultimate defeat of the Emperor and the Sith isn’t Rey becoming a Skywalker, but Rey bringing the name of Palpatine into the light as a Jedi instead of keeping the name in the shadows. (It also honors the sacrifice of her birth parents that died protecting her.)
From there, the stories write themselves: Jedi Rey Palpatine travels the Star Wars landscape helping others, rebuilding the Jedi Order, and working to undo the negativity of her family name while being tempted by the ghost of her grandfather. The remaining Imperialists (Thrawn?) want Rey to take her grandfather’s throne and lead them against the new(er) Republic. Scavengers, smugglers, and bounty hunters looking for Palpatine’s stolen wealth are chasing Rey planet to planet. And, real and wannabe Sith searches for the Emperor’s Darkside secrets – journals, laboratories, and weapons – and believe Rey knows where they are.
There’s your new trilogy, comics, books, television show, and games – the further adventures of Jedi Rey Palpatine.