Correct this if I'm wrong, but isn't the German eigen a contraction of ein and gehen? So it translates, elliptically as "one" with the infix, or inclusion, contracting to "gen", meaning "of/from-". And that means eigen simply means "from one (thing)"? This is extended to mean "having an equal outcome or probability of occurrence", when tacked on to the beginning of English words like "value", "function", ...
No. Eigen is descended from proto-Germanic agan, "to posess." It's the same word as English "own," with the standard phonetic shifts. I don't know German well enough to know whether the infinitive of the verb eigen still exists, but the adjective eigen is a corruption of its past participle and means "(one's) own." No again. The Wikipedia article on "Eigen..." states: Contractions are not as common even in colloquial German as they are in English. (E.g., Wie geht's? for Wie geht es (dir)?) In formal speech they're extremely rare.
