What is the significance of precisely 1,369 light bulbs used in the basement in the prologue?

The number can be significant in a few ways: first, it contains the first 3 multiples of 3: 3, 6, and 9.  Three is a significant number in Invisible Man: the narrator is the third generation since slavery was abolished (grandfather - father - son).  As you know, "eighty-seven years" is also mentioned often, the elapsed time since slavery.  1952 is the year of publication.  1952 - 87 is 1865, the end of the Civil War.  Ellison has great respect for that generation of freed slaves, and he urges his narrator and readers not to be ashamed of them.  So, he's asking, "it's been 3 generations, 87 years, of freedom and we're still invisible?"

Three is a magic number in the novel.  When the narrator is in the hospital "three hundred years" is mentioned.  Another time he says it took "three hundred years of black blood to build this white mahn's civilization."  The student body count is 3,000.  There are 300 teachers.  It's $300 rent.  Three fans.  Three old women.  Three girls.  Three hundred dollar bonus.  And so on.

Also, 1,369 is the only 4-digit square number that has all its numbers in succession.  The square root?  37.  That's the age of the author, Ralph Ellison, at the time he began the novel.  Succession is a motif in the novel: ("Keep that nigger boy running").  The narrator goes from the South to the North, from job to job, and from society to his hole--trying to find out who he is.  Eventually, he will come out.  So, the novel builds on this, like the number itself.

It's also two sets of numbers (if you keep the 1): 13 and 169.  13 times 13 equals 169.  Another editor has this to say (her math is a bit fuzzy):

The square root of 1,369 is 13, a prime number. Identifying 13 as a prime number is important: 13 is only divisible by itself indicating the only way to seek truth is through enlightenment. The analogy “hibernation” is a metaphor for the protagonist state of contemplation. Only through contemplation can the protagonist decide what to do when he awakens from his den.

And, I'm sure, it has a lot to do with music.  Some scale or time signature or jazz number...