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This has been bugging me for a while too... The only result I can find is the band 'Trotsky Icepick' which released two albums called 'Poison Summer' before finding a band name they wanted to settle on. Their biography's not on All Music Guide however, so I think a more recent band has come up with the same idea.
The K.L.F.? They changed their name tons of times in the early years.
This is ringing a bell, I think it was a fairly well-known band, but I can't remember which one.
The only thing I'm coming up with, and I don't think this is it, is that in the 1980s there was a band called Flowers whose debut album was called Icehouse, and so many people thought that it was an album called Flowers by a band called Icehouse that they changed their name.
Maybe I read about this on the other wiki instead of all music to begin with, because I looked up Trotsky Icepick
there and saw the following: "For their first albums (both entitled Poison Summer), they adhered to the concept that the name of the band would change with each new album while each record would bear the same title. With their third album ready for release the group finally bowed to record label pressure and permanently adopted Trotsky Icepick as a handle". Sounds to me like I've found the right band.
This is kind of a tough one, because what I remember isn't a song or music video, just an odd quirk of a band's discography: While doing a Wiki Walk through All Music Guide, I found the page of a band where their bio said that originally, the band had wanted to change their name every time they put out a new album, but always give their albums the same title (you know, the opposite of what bands normally do). After doing this for their first two albums, they decided it was a stupid idea and stuck to one band name from then on. Anyone know who this band might be?