What to give a one year old for a cough?

Vapor and love: There isn't really any medicine that's authorized for that age. Usually best thing is getting the bathroom steamy and sitting in there for half an hour or so. Helps the mucous drain. You can use Vicks (chest only, not on the feet), and nasal saline and suction to remove mucous from the nose. Other cold medicines are, unfortunately, off limits.