Buzz is the younger of the two children of Vinton Harper (whom he was named after) and his first wife, Mitzi. Right after Mitzi had run off to Las Vegas to work in show business (in actuality, she worked at a bar as a cocktail waitress) he, along with his father and his sister, Sonja Harper, moved in with his grandmother, Thelma Harper, after they were evicted from their house. He shared a very good relationship with his grandmother and great-aunt, Fran Crowley.
The reason for this is that he didn't constantly drive his grandmother crazy as Sonja was often wont to do, and as his father had to his mother when he was Buzz's age. He found favor with Aunt Fran because they shared a bathroom and at least, he was extremely neat and orderly, unlike Sonja, who strewed her makeup all over the counter.
The thing about Buzz was that he was extremely obedient, very patient, and was always willing to do the right thing, no matter what. In short, he was quite respectful to everyone. Although he could be head-strong, like most of his family, he was never outright mean.
This didn't please Sonja, who like her aunts before her, felt that he was overshadowing her. On the other hand, Buzz was himself, and it earned him favor from his great aunt and his grandmother.
Buzz also seemed to have a close bond with his grandmother. One example was when she was lining up help to clean the attic. After everyone else gave some ridiculous and even some not-so-subtle excuses for not wanting to do so, Buzz said that he had no excuse, and so said he would help her. "You've got no imagination, Buzz," his grandmother said, "but I love you."
Together the two shared stories from her earlier days when his dad and aunts were children, and even though they never got the attic (where Buzz slept) cleaned up, it mattered not; it was a fun time for both of them. As it turned out, it was a surprise for Thelma, who received a freezer that she had coveted for a long time, as a birthday present, and everyone had pitched in.
Another sore spot with Sonja was that even though she was older, she perceived Buzz to be favored by his father. In fact, he told his father, after he had told them that he was planning to remarry, that no matter what situations that he dragged them into, that they were his for the dragging. In his mind, other fathers would have given up on them and dumped them, but his father never did, and that was good enough for him.
Buzz and Sonja were on the receiving end of their sharp-tongued aunt, Eunice Higgins's ire when she told them to go "go soak their heads in Clearasil!!" This occurred at his father's wedding.
Buzz will defend his family, when push comes to shove. This was shown as Buzz had shouted at Eunice as she was being carried out by his Uncle Ed.
After two years of living with his grandmother, he and Sonja moved to an unknown location. They did establish a bond with their father's second wife, Naomi Oates Harper, however. In fact, Buzz outright said that he liked Naomi.