1580s, "a middle ground, quality, or degree," from Latin medium "the middle, midst, center; interval," noun use of neuter of adjective medius (from PIE root *medhyo- "middle"). Meaning "intermediate agency, channel of communication" is from c. 1600. That of "person who conveys spiritual messages" first recorded 1853, from notion of "substance through which something is conveyed." Artistic sense (oil, watercolors, etc.) is from 1854. Happy medium is the "golden mean," Horace's aurea mediocritas.
1660s, "average," from medium (n.). The Latin adjective was medius. Meaning "intermediate" is from 1796. As a size designation from 1711. as a designation of cooked meat, it is attested from 1931, short for medium-rare (1881).