Miyagi and Daniel's entire relationship, especially when you take their backstories into account. Daniel lost his father seemingly not long before the trilogy began, and Miyagi lost his wife and son while he was serving in World War II, so the two of them essentially are the family to each other that they couldn't have.
When Daniel asks what kind of belt Mr Miyagi has, the old man makes a joke about it being 'Canvas, JC Penney, $3.19' and how 'In Okinawa having belt mean no need rope to hold up pants' and laughs. Then...
Mr Miyagi: Daniel-san. Karate here [taps his head]. Karate here [taps his heart]. Karate never here [gestures to his belt]. You understand?
Kumiko, on Miyagi and Yukie's impromptu tea ceremony: "They're re-kindling their relationship, Daniel-san." And later Kumiko performs the same tea ceremony for Daniel.
Daniel begs Miyagi to let him come along when he goes to Okinawa to visit his dying father, on the grounds that, "When I need you, you're always there for me."
Airline Employee: Young man, the plane is boarding.
Miyagi[smiling]: Young man coming with us.
Later in the same movie, Daniel helps Miyagi mourn the loss of his father.
After the typhoon in which Sato learns the true heroism of Miyagi and Daniel, Sato comes with a work crew to rebuild the village and asks for Miyagi's forgiveness for all the trouble he caused. To that, Miyagi warmly responds, "Oh Sato, nothing to forgive."
During the typhoon, Daniel saved a little girl who got caught in the local tower during a typhoon which triggered both Sato's HeelFace Turn, going so far as to insist on helping Daniel, and his rejection of Chozen. Later, in a local celebration Daniel attempts to follow the little girl as she perfoms a traditional dance...
Daniel: If not for you, I'd have made a lot more mistakes.
Girl: If not for you, I no be here!
During Daniel and Chozen's final fight, and it looks like Daniel is gonna lose. Cue to Miyagi and the rest of the audience using little Japanese toy tambourines to show Daniel they support him. Daniel goes into My Name Is Inigo Montoya mode and wins. Mr. Miyagi's use of the drum wasn't just to show support, it was to remind Daniel of the drum technique he had taught him earlier, which Daniel promptly used to beat down Chozen. It's still heartwarming when you realize that, by reminding him of such a detail, he is supporting him in a way.
Daniel's birthday celebration at Mr. Miyagi's home in the first film was rife with this, as it showed that Daniel meant much more to Mr. Miyagi than just a student. When he took him to the driveway where he kept his collection of cars and simply said, "choose" the look on Daniel's face was nothing short of amazing.
Daniel(soon afterwards): You know, you're the best friend I've ever had.
Miyagi: You... pretty okay too.
While the end of the first movie was pretty heartwarming since Johnny gives Daniel the winning trophy personally. Props go to Miyagi for saving Johnny from Kreese in the second. Even though early on in the first he was the one fighting Johnny.
In the beginning of the second film, Mr. Miyagi has Daniel help him on an extension to the house. When Daniel asks him who it's for, he says a "refugee" from Fresno. He planned to let Daniel stay with him for two months while his mother took a business trip out of town.
A blink-and-you'll-miss-it moment in the third film (which overlaps with Tearjerker). Miyagi takes out a picture of himself and Daniel, who has been drifting away from him lately and has confessed that he feels 'confused.' Miyagi looks at the photo and softly utters two lines:
Miyagi: Hope confusion end soon, Daniel-san. Miyagi heart empty without you.
The ending of the third film, where Daniel hugs Miyagi happily. Also, before the final round, Daniel makes a point of bowing to Miyagi out of respect, and as an apology for being at odds with him during the event of the movie; Miyagi bows in return.
The first film, just seeing Miyagi beaming at his student's triumph is all the real reward Daniel needs and the perfect final image for the audience.
Miyagi waltzing with Julie, and later his pride at seeing her all grown up.