The Yardbirds’ manager, Giorgio Gomelsky, gave Eric Clapton the nickname “Slowhand” in early 1964.
The Yardbirds rhythm guitarist, Chris Dreja, recalled that whenever Eric Clapton broke a guitar string during a concert, Eric would stay on stage and replace it. The English audiences would wait out the delay by doing a “slow handclap”. [The British colloquialism is "to be given the slowhand".]
Clapton told his official biographer, Ray Coleman, in the mid-80s that “My nickname of 'Slowhand' came from Giorgio Gomelsky. He coined it as a good pun. He kept saying I was a fast player, so he put together the slow handclap phrase into 'Slowhand' as a play on words.”
In a June 1999 online chat, Clapton gave a slightly different version of how his nickname came about: “I think it might have been a play on words from the “Clap” part of my name. In England, in sport, if the crowd is getting anxious, we have a slow handclap, which indicates boredom or frustration. But it wasn’t my idea it was someone else’s comment.”
In Clapton - The Autobiography (2007), Eric had this to say, "On my guitar I used light-gauge guitar strings, with a very thin first string, which made it easier to bend the notes, and it was not uncommon during the most frenetic bits of playing for me to break at least one string. During the pause while I was changing my string, the frenzied audience would often break into a slow handclap, inspiring Giorgio to dream up the nickname of 'Slowhand' Clapton."