Zarema is a modern Russian name. No meaning was given for it at the Russian language baby names site I found it at; as far as I can tell from further searching, it’s of Chechen origin. The one site which gave a meaning said it may mean “sweet water” or “war maid.” This name is also possibly derived from the Persian name Zareen, meaning “golden.”
I personally tread very carefully when taking any name site besides Behind the Name as an accurate source for name meanings. It’s all about vetting your sources. I’ve found out a lot of names don’t have the meanings I was led to believe they had. Honestly, a lot of name sites are garbage, the way they lump names in categories they clearly aren’t part of, and by giving blatantly untrue meanings and etymologies. Zarema is obviously a very real name, but I’m not going to definitively give it a meaning or etymology in the absence of scholarly sources.
Zoriy is a modern Russian name, not an invented Soviet name. It means “morning” in the adjectival form. Russian is such an amazingly rich language, with so many forms of words branching off from one simple root. One of the basic nickname forms would be Zorik.
http://panzercentral.com/forum/viewtopic.php?f=96&t=36301 (penultimate post)