c. 1200, "either, else, otherwise, as an alternative or substitute," from Old English conjunction oþþe "either, or," which is related to Old Frisian ieftha, Middle Dutch ofte, Old Norse eða, Old High German odar, German oder, Gothic aiþþau "or."
This word was extended in early Middle English (and Old High German) with an -r ending, perhaps by analogy with "choice between alternative" words that ended thus (such as either, whether); then it was reduced to oþþr, at first in unstressed positions (commonly thus in Northern and Midlands English by 1300), and finally to or, though other survived in this sense until 16c.
Compare either, which is originally the same word. The contraction took place in the second term of an alternative, such as either ... or, descended from a common construction in Old English, where both words originally were oþþe (see nor). Or else "otherwise" is by c. 1300.