Soloon was one of three brothers (the others were Euneüs and Thoas). He was a companion of Theseus when he visited the Amazons. Having fallen in love with Antiope, but not daring to approach her personally, he drowned himself in the river Thermodon out of unrequited love.
His name seems to be a blend of solos and oon, 'egg'. The solos was a mass of iron used as a weight for throwing in games, like a discus but spherical. So he was, or he represented, an 'egg-weight'! This presumably refers to some sporting contest rather than being a personal Humpty Dumpty-style epithet.