Let me start by saying, I like Carrot. She’s quirky, strong, knows when to be serious vs silly, and has a fun dynamic with the rest of the Straw Hats. All the ingredients needed for a new crew member.
However, when people argue for this they usually start with the conclusion (“I like Carrot, she’s joining the Straw Hats) and then search for evidence to prove themselves right. THIS IS BAD SCIENCE.
So instead, let’s be good scientists and look at all the evidence first, then see what conclusions we can draw. Below are quick summaries of each Straw Hat’s journey into the crew. Let’s see if we can find any patterns that point to Carrot being on this path.
Zoro – Luffy says he wants Zoro to join his crew in Chapter 2, before they’ve even met. They meet in Chapter 3 and Luffy insists he joins. Zoro declines. After fighting together through out the arc he accepts.
Nami – Luffy meets Nami in Chapter 8, at the end of which Nami proposes they team up. In Chapter 9 Luffy formally asks her to join the crew. She declines. She then tricks Luffy by saying she’ll join if he helps her steal from Buggy. He agrees. They sail and fight together through out the next four arcs. At the conclusion of Arlong Park she officially becomes a member.
Ussop – Luffy meets Ussop in Chapter 23, at the end of which Ussop suggests he joins the SHs, as captain. Luffy declines. After fighting together through out the arc Usopp joins (not as captain). Luffy says “you’re already our crew mate, aren’t you?”
Sanji – Luffy meets Sanji in Chapter 44. At the beginning of Chapter 45 Luffy asks Sanji to join his crew. Sanji declines. After fighting together through out the arc Sanji accepts.
Chopper – Luffy meets Chopper in Chapter 139. He tries to eat him. In chapter 140 Nami asks Chopper to join the crew. He declines. At the end of that chapter Luffy asks Chopper to join. He declines again. After fighting together through out the arc Chopper accepts.
Robin – Luffy meets Robin in Chapter 114. He can’t decide if she’s good or bad. They don’t see each other again till Chapter 170. Robin saves Luffy’s life in Chapter 180. Robin turns on Crocodile in chapter 203. Robin saves Luffy’s life from Croc’s poison, Luffy saves Robin’s life from collapsing building. After the events of Alabasta Robin sneaks on board the Merry and asks to join. Luffy says yes, concluding that she in fact good.
Franky – Luffy meets Franky in Chapter 335. They hate each other and fight immediately. They later join forces. After fighting together through out the arc Luffy asks Franky to join. Franky at first declines, but after heartfelt speeches from his friends (and Robin’s, ahem, persuasive hands) he accepts.
Brook – Luffy meets Brook in Chapter 442. He asks him about his poop then asks him to join his crew. Brook accepts but later declines. After fighting together through out the arc Brook accepts.
Jinbe – Luffy meets Jinbe in chapter 540, Ace had already told Jimbei all about Luffy and asked him to protect him. Jinbe carefully analyzes Luffy to see if he’s worthy. They fight and travel together for the next two arcs. Jinbe saves Luffy’s life a number of times and councils him through his grief. They separate at chapter 597. They meet again (after the timeskip) in Chapter 619. After fighting together through out the arc Luffy asks Jinbe to join his crew. Jinbe reluctantly declines. They meet again in Whole Cake Island and in chapter 863 Jimbei officially declares his intention of joining the Straw Hats. He doesn’t get his own “Member of the Straw Hat Pirates” text box until the most recent chapter, 977
So what have we learned?
A pre-requisite to join the crew is fighting alongside the Straw Hats through out an arc. Check for Carrot!
However, the best bet in determining if someone’s going to become a Straw Hat is if LUFFY ASKS YOU TO JOIN RIGHT AWAY.