Symbolism, Imagery, Allegory
Rule of thumb: when anyone in literature (or, hey, in real life) mentions that they'd like to be a bird, there's an approximately 99% chance that they're wishing for a little more freedom.
In Book One: Fear, Bigger and Gus "play white," demonstrating how trapped they feel in their own lives and how much comparative freedom white people have. When they finish, the young men watch as a pigeon lands on the cable car tracks and struts around, then flies away as a street car approaches:
Then their eyes were riveted; a slate-colored pigeon swooped down to the middle of the steel car tracks and began strutting to and fro with ruffled feathers, its fat neck bobbing with regal pride. A street car rumbled forward and the pigeon rose swiftly through the air on wings stretched so taut and sheer that Bigger could see the gold of the sun through their translucent tips. He tilted his head and watched the slate-colored bird flap and wheel out of sight over the edge of a high roof.
"Now, if I could only do that," Bigger said. (1.292)
The pigeon represents freedom, the ability to go where he wants when he wants, instead of being stuck where he is.