I just learned yesterday that the lyrics were written by the son of Robert Altman (the director of the movie M*A*S*H), when he was 14 years old.

From Wikipedia:

The song was written specifically for Ken Prymus (the actor playing Private Seidman), who sang it during the faux suicide of Walter "Painless Pole" Waldowski (John Schuck) in the film's "Last Supper" scene.[1][2] Robert Altman had two stipulations about the song for Mandel: first, it had to be called "Suicide Is Painless"; second, it had to be the "stupidest song ever written".[3] Altman tried to write the lyrics himself, but found that it was too difficult for his 45-year-old brain to write "stupid enough".[4] Instead he gave the task to his 14-year-old-son, Michael, who apparently wrote the lyrics in five minutes.[5][6]

I don't think they succeeded in making the stupidest song ever written.

I just found a great interview with Mandel:

JW: How did you work on the project?

JM: I was brought in before the movie was even shot, which was highly unusual. In most cases, you’re the last one in the line to see the film when scoring it. So Bob and I were sitting around getting rather ripped one night. Bob said to me, "You know, I need a song for the film. It’s that Last Supper scene, after the guy says he’d going to do himself with a pill because his life is over, because couldn’t get it up with the WAC the night before." I said, "A song for that?" He said, "Yeah, that Last Supper scene where the guy climbs into the casket and everybody walks around the box dropping in things like scotch, Playboy and other stuff to see him into the next world. There’s just dead air there."

JW: But if I recall, the scene features just a guy singing with an acoustic guitar.

JM: Right. Bob said, "We’ve got one guy in the shot who can sing and there's another guy who knows three chords on the guitar so we can’t use an orchestra." Bob also said the song had to be called Suicide Is Painless. "Since [Capt.] Painless commits suicide with a pill, that would be a good title," he said. Then he said, "It’s got to be the stupidest song ever written."

JW: What went through your mind?

JM: I said to myself, "Well I can do stupid." Bob was going to take a shot at the lyrics. But he came back two days later and said, "I’m sorry but there’s just too much stuff in this 45-year-old brain. I can’t write anything nearly as stupid as what we need."

JW: So who wrote the lyrics?

JM: Bob said, "All is not lost. I’ve got a 15-year-old kid who’s a total idiot." So Michael Altman, at age 15, wrote the lyrics, and then I wrote the music to them. It was the first scene in the movie that they were going to shoot. They had to have the song for it as a pre-record, so the actor could mouth the words, allowing for a dub later.