mid-13c., dremen, "to have a dream or dreams, be partly and confusedly aware of images and thoughts during sleep," from dream (n.). Transitive sense of "see in a dream" is from c. 1300. Sense of "think about idly, vainly, or fancifully; give way to visionary expectation" is from late 14c. Related: Dreamed; dreaming. To dream up "picture (something) in one's mind" is by 1941.
In the older sense of "sing, rejoice, play music," it is from Old English drēman (Anglian); dryman (West Saxon), from the Old English noun. This was obsolete from c. 1300.