For the multitude of tearjerkers in Hard Luck Woman, the way that Ein joins Ed is very heartwarming. Ein wakes up, seemingly in reaction to Jet's cooking, but then he ignores the food and runs out of the ship to catch up with Ed, who the viewers don't realize is leaving yet. Outside the ship, he wavers for a second, and obviously considers going back inside, where he'll be certain to be fed and at least somewhat taken care of. But instead, he keeps going and catches up with Ed, so that they'll be together, whatever happens. A girl and her dog, indeed.
Everyone on the crew seems to like Ed, especially Jet. Faye encourages Ed to look for her father, Jet tells her stories and Spike even gives her the cowboy hat he earned from Andy, not to mention the look on his face when she gives him a paper pinwheel in episode 24. Never mind what he does with the pinwheel itself: He tapes it to the bow of the Bebop.
Not to mention the time where they gave up a huge bounty just so Ed could finish her chess game with that week's criminal. 'Course, they were doing it for the criminal too, because they felt him going to jail was too harsh considering he was too senile to remember what he did.
The male co-host of the bounty hunter program getting a sweet moment with his mother in the finale.
The Photograph at the end of "Wild Horses".
The entire conflict in "Brain Scratch Fever" is based around the rest of the Bebop crew trying to rescue Faye from a cult that has brainwashed her. Spike and Jet had issues with her at the start, but at this point they definitely consider her a true companion.
A Meta example: Every fan lives for the words "See you space cowboy".
In a weird way, when Chessmaster Hex peacefully passes away at the end of Bohemian Rhapsody. There's something comforting about knowing that despite everything that happened to him and that he'd done, Hex was granted a peaceful death, having just beat Ed and calmly letting go.