The common noun Debbie Downer denotes a person who is habitually pessimistic, negative, gloomy, etc.

It was popularised, if not introduced, by the character of Debbie Downer—whose distinguishing trait are her negative pronouncements—in the U.S. television variety series Saturday Night Live.

The character’s surname refers in turn to the common noun downer, denoting someone or something depressing, disagreeable or unsatisfactory.

The character of Debbie Downer, portrayed by the U.S. comedian Rachel Dratch (born 1966), first appeared in Saturday Night Live on Saturday 1^st May 2004. David Bianculli reviewed that show in Cracking up ‘Live’ on air, published in the Daily News (New York City, N.Y.) of Tuesday 4^th May 2004:

A very funny thing happened on last weekend’s live telecast of “Saturday Night Live” — but it wasn’t what you would expect.
Fact is, it wasn’t even planned.
That’s because during one skit, ironically titled “Debbie Downer,” the entire cast fell victim to an infectious giggle fit.
[…]
The previous hour of that “SNL” was building, slowly but surely, to a giggle fit. […]
When “Debbie Downer” came around, there was no stopping it.
The sketch took place at a theme eatery at Walt Disney World, with Jimmy Fallon, Rachel Dratch, Amy Poehler, Logan, Horatio Sanz and featured player Fred Armisen at a large round table. Whatever the topic of conversation, it would soon turn to Dratch’s Debbie, who would spout some depressing comment, then stare into the camera for an extreme closeup, holding a sad face as the band played a cartoon-style trombone waaah-waaaah sound.
The first time the cast tried it, it worked fine. The second time, right after the cartoon sound, Fallon cracked up. Lohan and Sanz fought to keep a straight face, but Dratch flubbed her next line (“The media are so sensitive there,” she said of Korea, before correcting it to “secretive”), and that was that.
Fallon was gone. Dratch, gone. Poehler, gone. Lohan, gone. Sanz was on the edge — but once Fallon noticed that, and started stuffing his mouth with a ridiculous amount of food, Sanz was gone, too, wiping his tears away with his shirt sleeve — then with a pancake.
Armisen, the least familiar of the group, earned his stripes by hanging on the longest, but eventually, the nationally televised giggle fit claimed him, too. Somehow, Dratch made it to the finish line, alone on stage to deliver her last downer line.

This photograph from Saturday Night Live of 1^st May 2004 was published in the Daily News (New York City, N.Y.) of 4^th May 2004:

BIT PLAYERS: Fallon (from l.), Dratch, Poehler, Armisen, Lohan & Sanz on glee-filled “SNL”

On 26^th August 2004, during The Oprah Winfrey Show, the U.S. actress, singer, author and businesswoman Gwyneth Paltrow (born 1972) apparently identified herself with the character—source: Oxford English Dictionary (3^rd edition, 2018):

I’m Debbie Downer over here.

The earliest form of the common noun that I have found may have been coined independently from the character’s name in Saturday Night Live, since it is Debbie-the-Downer. It appeared in Tsunami victims need help from all who can give, by Annamaria Longo, published in the El Paso Times (El Paso, Texas) of Saturday 1^st January 2005:

I don’t know about you, but natural disasters scare the living daylights out of me. […]
Earthquakes, tsunamis, hurricanes, typhoons, tornadoes, snowstorms — they aren’t human. We can’t reason with them. No matter how many warning systems we have in place, we can’t control them.
Now, I don’t mean to be a Debbie-the-Downer. It’s a new year, after all. A clean slate full of promise and expectations. But — and this is a very big but — sometimes things affect you in such a way that you have to tell somebody. And today that somebody is you.

The earliest instance that I have found of the common noun in the form Debbie Downer is from Here comes some trouble, in the section Go! Weekend of The Boston Globe (Boston, Massachusetts) of Friday 4^th March 2005:

Few bands unsettle Go! quite as much as Shivaree¹. Not that there’s anything wrong with that. In Shivaree’s case, we can’t quite figure out what’s going on with the bewitching lead singer, Ambrosia Parsley² […]. On one song, she’s a chirpy alt-country vixen, the next a humdrum Debbie Downer, and then, on another, Suzanne Vega³’s doppelganger.

¹ Shivaree is a U.S. band formed in 1997.
² Ambrosia Parsley (born 1971) is a U.S. alternative pop/rock singer-songwriter.
³ Suzanne Vega (born 1959) is a U.S. folk-inspired singer-songwriter and musician.