What does "omenis" mean?

"Omenis" [ɔ:mɛnɪs] is an archaic Lithuanian word for "mind" in somewhat narrower, more physical meaning, for example:

Lithuanian: "Turėk omeny, ką jis sakė, tai turėtų padėti"
English key: "Have in [your] omenis, what he said, it should help"
English: "It should be helpful, if you would be prepared to remember and act according to what he said"

It feels like any concept is expected to work simply because its location in omenis. If it doesn't work - you missed (the point) and the concept was left outside your omenis. The most exact, shortest English translation, that I can think of - "remembrance capacity".

To be exhaustive about Lithuanian language, there is one more, casual and truly rare use of "omenis" for "consciousness". One may think of unconscious man as of a man, who (temporarily) lost his "remembrance capacity".

Lithuanians frequently use "omenis" in their everyday speech, but there is a catch - they use it almost exclusively in locative case "omenyje", "omeny" [ɔ:mɛnĩ:] (as it was in the example above).

Lithuanian dictionary 

Is omenis.net just a personal homepage?

Lets say - it is under construction. The construction itself is not to be disclosed yet.

Is it OK to write Juzikenas instead of Juzikėnas ?

Yes, it is OK for non-Lihuanian texts. Anyway, such minor diacritics won't help much for foreigners to pronounce my name [lɪʊtɑ:uɹas jʊzɪkenas].