Where does the phrase "on end" come from?

The phrase "on end" means "without end". It very much sounds like the German "ohn End" which itself is the short form of "ohne Ende".

Is this etymologically the right direction? (Sometimes these similarities are misleading!)

Since nobody has answered this yet, here's my formalization of the comments. "On end" means "without a break", which has been slightly altered over time so that "days on end" now often means "several days". It is easy to misunderstand this as "days without end", and to wonder about the origin of the phrase. But if you do this, you are committing a solecism of which we are all guilty from time to time; assuming that something you do not know must be related to something you do know. (Is there a name for this, all you linguists?)

So technically the answer is that your questions is based on a false premise, and cannot be answered.