Where does the phrase "to wash up" (equally "to clean up") originate from?
Particularly the word "up", how did that enter the phrase?
Both wash up and clean up are phrasal verbs, which are very common in English. These verbs then to share a common etymology: old english verbs with separable preposition prefixes. Suite101 has a nice write-up:
Phrasal verbs in Modern English developed from verbs with separable preposition prefixes in Old English. Verbs with separable preposition prefixes still exist in Modern German and Modern Dutch. For example, the Modern German verb aufwachen "to wake up" consists of the verb wachen and the preposition prefix auf. When the verb is conjugated, the preposition prefix moves to the end of the predicate phrase as in ich wache auf "I wake up." Old English verbs with separable preposition prefixes evolved into phrasal verbs in which the preposition follows the verb in Middle English.
There are many phrasal verbs in English, consisting of a verb plus an adverb or preposition, and for most of them the meaning cannot be deduced from the component words.
There are plenty in "up": "give up" (= "surrender"), "pull up" = "stop moving", "wait up" = "not go to bed while waiting for something".
"Up" in such a phrase often has a connotation of finishing or completing something, but I can't see much of that meaning particularly in this phrase.
"Wash up" actually has a significantly different meaning when you cross the Atlantic. In the UK it means "wash the dishes"; I believe that in the US it means "wash one's hands". I don't know what it means in Canada.
"Up" can also be an adverb to mean completely or thoroughly. Therefore "wash up" means to completely wash. Using "up" like this is actually pretty common, but you don't notice it much:
I could beat him up.
He forgot to put them away separately, and now they're all mixed up.
Oops! I messed up!
I have to type up the paper tonight.
My dog tore it up yesterday.
Go wash up.