In "Master and Margarita," why does the master have no name?
The symbolism of a "generic" name such as "The Master" is that it represents a whole class of people. The Master has no name because he is a figure who is intended to represent any true artist who undergoes trials and persecutions in the cause of his art.
In being symbolic, "The Master" is only one example of Bulgakov using symbolic or allusive naming in his novel. The name of the Master's mistress and disciple, Margerita, alludes to the Faust legend, as does the nom de guerre the Devil adopts, Woland (mentioned by Goethe in Faust). The poet who becomes another follower of the Master, Ivan Bezdomniy, might be more literally called "Ivan Homeless." The Devil's followers also have names with multiple layers of meaning: the cat Behemoth alludes to the Biblical monster as well as evoking the Russian word for hippopotamus; Azazello reminds us of Azrael, another demon; while the references in the names of Abadonna and Hella are obvious. These multiple examples make it clear that the generic labeling of "The Master" was intended to evoke something beyond its surface meaning.
I'm convinced that he does have a name, however, because his life is defined by Margarita, he introduces himself as "the Master." If you notice, when he introduces himself to Ivan, he says, "they call me The Master." And then to prove that he is the Master, he puts on a cap made by Margarita. Margarita controls the Master so much, that he loses all identity with exception to what she had called him. It was Margarita who pushed the Master to write his novel, and it was her who saved it from the flames, it was her action that gives any substance to the character of "the Master."