Why is young Gillian called the "black sheep" of the family?

Young Gillian actually labels himself the "black sheep" because he does not value money, nor does he work; instead, he lives off the allowance that his uncle affords him.

After his uncle dies, the law firm of Tolman & Sharp summons Gillian in order to provide him $1000 with the instructions that he must spend it and provide an accounting for this expenditure. Gillian finds the money "a confoundedly awkward amount" and has difficulty deciding how to spend it. So, after asking Old Bryson at the men's club, a cab driver, and a blind man, as well as suggesting a diamond pendant for his favorite actress, Gillian has but one last idea. With this he hurries to his uncle's house where his ward, Miss Hayden, resides. 

When he enters the library in which Miss Hayden sits writing letters, he tells her that his uncle had a codicil to his will in which he willed her a thousand dollars. Since he "was driving up" anyway, he explains, he decided to bring it to her. "Oh!" Miss Hayden exclaims in surprise, and Gillian declares his love for her. But, as she takes the money, Miss Hayden declines his declaration of love, saying, "I'm sorry." So, Gillian asks if he can write a note, and he retires to a desk in order to write out his accounting of the expenditure of the $1000:

Paid by the black sheep, Robert Gillian, $1000 on account of the eternal happiness, owed by Heaven to the best and dearest woman on Earth.

With this act of unselfish love, Gillian proves the irony of his being termed a "black sheep." Further, when he learns that he will receive more money for having dispensed of the $1000 in an altruistic gesture, Gillian tears up this note so that he will have none of "the qualifications that deserve reward" and thereby retain his truly undeserved label of "black sheep."

This ironic reversal to O. Henry's story demonstrates that there is only superficial truth to the reputation Gillian has of being a black sheep, and not caring about how much of his uncle's money he spends. Rather, he simply does not care whether he has money or not; he is completely unmaterialistic.