I'd like to know the exact answer for that as well. It is a great place, and has so many assets that make the place so unique and special among other cities; especially those of similar size. But I know what people mean when they call Cleveland this. You see, there are people who will answer your question by pointing out all the misfortunes in the city's history that goes back over a hundred years. The misfortunes they might include might be the decrepidation of Euclid Avenue, the departure of the Rockefellers and other industrialists from the area, urban sprawl, some terrible political corruption, pollution, racial issues, unsuccessful sport teams, the Cuyahoga river catching on fire, the loss of Fortune 500 companies BP America headquarters and TRW, the end of the Cleveland Ballet, brain drain, unemployment and poverty statistics.
I don't know when this nickname originated, but it probably came about during the fire on the Cuyahoga in 1969 that made national headlines and was highly publicized. Though historians now look back on the incident as major media hype (who determined the fire was neither severe, ecologically damaging, nor unique to Cleveland), it was the source of great embarrassment and frustration for local Northeast Ohio residents who continue to believe Cleveland should be ashamed of itself. Since then, bad news and bad reputations always takes precedent over good news and good reputations in Cleveland.
I think this title and more generally speaking the culture of negativity in Cleveland is based subconsciously on the fact that Cleveland was for a brief period of time the twenties the most prosperous and prestigious city in perhaps the whole world and lost it practically over night. The "mistake on the lake" alludes to and replaced the former slogan of the city as "the best location in the nation". Cleveland's location as lying in the center of important crossroads as well as lying on Lake Erie and the terminus of the Ohio Erie Canal led to its prosperity. Look at a map and you will see Cleveland is the only big city between Cincinnati and Buffalo, Detroit and Pittsburg, and New York and Chicago. Before the Great Depression Cleveland was the sixth largest city in the country, claimed the most millionaires per capita in the world, was said to be the country's second most important international commercial center, had extremely exclusive stores and restaurants, and Euclid Avenue was considered the most beautiful street in the world.
Cleveland lost so much of its status when the Great Depression hit, and since then it has tried to reclaim its glory and rebuild itself but terrible things always kept happening. People have the feeling that the city's glory days were a fluke, and the city should have never been founded at all based on all kinds of bad news. The weather doesn't help either. There are so many reasons to be proud of the city, but most people aren’t aware of any. I could go on and on and count these numerous reasons that give testament to the reality of Cleveland's first class status, but that isn't necessarily relative to your question. If you want more information about why Cleveland is undeserving of that title today, just let me know.