Firstly, according to Jeff Chang's pretty definitive Can't Stop Won't Stop: A History Of The Hip-Hop Generation, hip-hop evolved out of a collision between Jamaican sound system culture, where DJs toasted over records, and 1970s African-American culture in the Bronx. Chang calls it a 'creation myth' that the genesis of hip-hop was a house party in 1973 where DJ Kool Herc - an aspiring DJ who had grown up in Jamaica before moving to New York - was the DJ at a party put on by his sister Cindy, where he initially was playing Jamaican music, but discovered that the crowd wanted soul and funk. He thus played a set focused on soul breaks, through his powerful Jamaican-style sound system (which were renowned for their deep bass response), and word of mouth quickly spread.

Based on the success of this party, Kool Herc would soon develop a set based around looping 'breaks' (i.e., the small drum solos in soul funk tracks where the drummer showed their stuff, like the very popular one in James Brown's 'Funky Drummer'). This quickly was very popular, and by 1977 Kool Herc was one of the most popular DJs in the Bronx and had already inspired Afrikaa Bambaataa to emulate his style. Similarly, by 1977, Grandmaster Flash (who had started out in 1975 after studying Kool Herc and thinking he could do better) had a posse of rappers (which was originally the Three MCs but soon developed into the Furious Five), and had become the most popular DJ in some areas of the Bronx. By this point, most DJs in the scene had MCs who would sporadically rap over the beats and breaks.

Which is to say that, by the 1977 New York City blackout, hip-hop was already well-known in the African-American community in New York. The Jeff Chang book mentions the 1977 blackout in detail, but more to set up the context of how desperately poor and left-alone the Bronx was; he paints the looting which occurred during the blackout as a way to demonstrate the results of that poverty.

Chang does point out that after the blackout, there was an influx of new crews with sound systems:

By the end of the spring [of 1977], Herc noticed his audiences were declining. “People are getting older now, it wasn’t all about me. All of a sudden now you’re not eighteen no more, you’re twenty-four and twenty-five. You can drink now. You ain’t coming to no little seventeen-, eighteen-year-old party,” he recalls. “And other people was coming up.”

After the blackout and the looting, there were plenty of new crews with brand new sound systems in the streets, and Herc’s main rivals were luring away his crowd. Flash had precision, sophistication and an entertainer’s flair. Bambaataa had his records and the power of Bronx River behind him.

Similarly, in a well-researched 2016 Guardian article by Dorian Lynskey, which interviewed Grandmaster Flash, there's a quote from Flash about the blackout.

Flash was at home on Fox Street. “Shit was fucked. You could see people walking by with mattresses on their heads. You could hear glass breaking.” Some local looters presented him with a high-end stereo system. “I said, ‘Where y’all get that from?’,” he remembers. “‘None of your business. Now when you come out into the street you gonna sound good.’”

Lynskey also interviews Nelson George, who was the first person to write about hip-hop in Billboard magazine in 1979. Nelson's book Hip Hop America is a slightly more personal history of hip-hop than Chang's, and Nelson was a consultant on the recent fictional Netflix series The Get Down, which had an episode centered around the blackout:

“The blackout had a huge impact on hip-hop,” says George. “After the riot, there were suddenly a million crews with stolen turntables.”

While this is obvious hyperbole, the popularity of hip-hop in the Bronx by 1977 meant there were going to be people who aspired to be DJs but who didn't have access to the equipment necessary to do so effectively. So the blackout might have helped give hip-hop enough critical mass to be able to exploit the public's interest in the wake of the success of 'Rapper's Delight' by the Sugar Hill Gang in 1979.

The blackout also, according to Chang, helped make the wider public realise the poor state of the Bronx; in the wake of the blackout, Bill Moyers at CBS devoted a program to the poverty of the Bronx, and President Jimmy Carter rocked up at Charlotte Street in the South Bronx, looked around and said "see what areas can be salvaged." The coverage of the blackout might have played a role in people outside of the hip-hop culture and the Bronx becoming aware that it existed; Chang quotes a Long Island graphic design student called Carlton Ridenhour - now better known as Public Enemy's Chuck D - as saying he was inspired to start writing rhymes in 1977 after the blackout.