Well, there are competing theories, none of which is all that convincing but which are fun. This happens with local names and such. It is interesting that there was a London slum also called Hell's Kitchen, which is attested earlier. Perhaps someone borrowed the name. (Though independent invention is always possible.) Another says that the name originated with a single building, then grew to encompass the whole area. We do know that the nickname was first referenced in print in 1881, in the New York Times, referring to a tenement at 39th Street and 10th Avenue.
A particularly colorful story, which I like a lot, is that a local cop called Dutch Fred was showing a rookie around, observing an altercation. The rookie said that the neighborhood was Hell itself, to which Dutch Fred replied "Hell's a mild climate. This is Hell's kitchen."
This is from the Hell's Kitchen neighborhood association. This tells the story of Dutch Fred the Cop.
As a resident of Hell's Kitchen, I have read up on these theories, none of which seem definitive, but all are interesting.
In "Naming New York: Manhattan Places & How They Got Their Names," by Sanna Firestein, the was a park called Hell's Kitchen Park in the 19th century which later became the name of gang and later the name of a neighborhood.
In "The Encyclopedia of New York City," by Kenneth Jackson the name was either the name of a local gang or it was name given to the neighborhood by local police "this is Hell's Kitchen.." for example.
Another theory is that it was the name of a tenement or a slum on 39th and 10th Avenue before it became the name of a neighborhood.
Theodore Drieser writes about Hell's Kitchen in the 1890s as a place of unimaginable danger to someone who lives there today. It was a place of such ill-repute that legend has it residents would throw stones at strangers and sometimes would invite strangers in to drink and then douse them with flammable liquids and set them aflame (this seems unwise, you'd only have to set one person on fire for people to get the message not to accept a drink inside the home of a Hell's Kitchen resident). In Drieser's book, The Color of a Great City, Hell's Kitchen is an area bounded by 36th street and 41st street between Ninth Avenue and the Hudson River,"[I]t was a whim of the New York newspapers to dub that region on the West Side which lies between thirty-sixth and forty-first streets and Ninth Avenue and the Hudson River as Hell's Kitchen." Drieser wrote.
What is definitely true is that Hell's Kitchen was the site of some of the most serious race riots in NYC history. The famous Civil War draft riots of 1863 took place in part on 8th Avenue between 39th and 41st street where many Irish workers worked in a foundry. Major Race riots broke out in Hell's Kitchen in 1899 and 1900. In fact, the racial strife between the black residents of Hell's Kitchen and recent immigrants were so bad, black residents started migrating to Harlem, which became a largely black neighborhood by the 1920s. For more information read: Before Harlem: The Black Experience in New York City Before World War I by Marcy S. Sacks.