In the movie, the reason the Maitlands decide to let the Deetzes stay? Because they like Lydia. So in the end, the Maitlands are acting practically like a second set of parents and use their ghostly powers to play music for her is just pure d'aww.
When Lydia tells Beetlejuice she wants to die, he is actually concerned. He keeps Lydia from leaving the house, intending to talk her out of it (although he is manipulating her at the same time).
Beetlejuice: You know, you look like somebody I can relate to. Maybe you could help me get out of here, you know, because I got to tell you, this dead thing... it's just too creepy. See, here's my problem. I got these friends I said I'd... I have to be there in person, so could you help me get out of here?
Lydia: I want to get in.
Beetlejuice:(genuinely surprised) Why? (pause) You know, hey, you probably got your reasons. I can't do anything from here. If you could get me out, then maybe we could talk.
The Animated Series
In the cartoon, pretty much any moment shared between BJ and Lydia, especially if BJ is helping her out. Their entire relationship is just one big Crowning Friendship of Heartwarming. Highlights include:
In "Spooky Tree", Lydia chains herself to a tree to prevent it from being chopped down. When she gets tired, Beetlejuice offers to take over for her. It's small, but considering that earlier BJ said he didn't care what happened to the tree, it says a lot.
Then BJ goes out of his way even further in making the tree come to life, so it can move itself and be saved. The tree causes him a lot of trouble, but BJ keeps on going - all for Lydia.
Beetlejuice gets split in two during the episode "Spitting Image", and if it's not heartwarming enough that both sides fight for Lydia's attention, both sides (having fought for most of the episode) decide in the end to merge back together, in order to make Lydia happy again. She hugs him in response.
"Worm Welcome". Just look at the expression on his face when he realizes Lydia's in trouble, and even helps her save a baby sandworm, creatures he hates above all else.
In "It's A Wonderful Afterlife", Beetlejuice wishes away his life, and while everyone in the Neitherworld seems happy that he's not around, Lydia is very miserable. Just seeing her like this - and realizing he can never see her again - makes Beetlejuice want his old life back. And he was pretty much OK with everything else up until that point.
The Robin Hood episode shows Beetlejuice as his greedy old self, wanting to keep the money he stole. However, when Lydia is kidnapped, Beetlejuice is willing to give up this same money - yes, all of it - in order to find Lydia.
"Pest O' The West". BJ is on the fence about taking on Bully the Crud - the moment Bully makes a move towards Lydia, BJ gets serious.
BJ then pulls a Big Damn Heroes in storming into the wedding between Bully and Lydia (unwilling on her part), grabbing Lydia and getting her as far away from Bully as possible, placing himself in front of her. Kind of ironic when you consider that he himself was going to do the same thing as Bully in the movie...
"Out Of My Mind" reveals that Beetlejuice has a huge shrine to Lydia in his head. It's heartwarming...and a little creepy.
Notice that whenever Beetlejuice pulls a prank on Lydia, it's harmless and borders more on teasing, which they both just laugh off. This is common in any friendship, but remember that this is Beetlejuice we're talking about here.
The episode "Ghost To Ghost" reveals that despite his various pranks on them, Beetlejuice has a soft spot for Lydia's parents.
At one point an evil spirit is set loose that can only be defeated by telling the truth. Lydia begins a litany of questions for Beetlejuice to answer truthfully beginning with: "who's your best friend?" His immediate response: "You are."