"Rina" could be an abbreviation of the Bulgarian name "Zaharina" (the stress - on the "i"), coming from the word "zahar" (the stress - on the first "a"), which means "sugar" as a noun in Bulgarian. It could also be an abbreviation of the Bulgarian name "Zarina" (the stress - on the "i"), which is a derivative of "Zara".
Another name, which ends in "rina", is the Bulgarian feminine name "Dobrina" (the stress is on the "i"). The Bulgarian word "dobrina" (the stress is on the "a") means "goodness", "good nature".
Another name ending in "rina" is the Bulgarian female one - "Darina", but its meaning is connected with the word "dar", which means "gift" in Bulgarian and because Jesus Christ is "a Gift of God"; in Bulgaria "Darina" and all other names beginning with "dar" have a nameday on 25th December.
My best friend's name is Terina and I call her Rina for short. It is very pretty and girlie. I think it is a lovely name for a person who is loved by all and very friendly!
― Anonymous User 7/10/2005
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But what / how many other names end in -rina besides all the various forms of Jekaterina through to Katharina? That is the most common and obvious source.