Russians used to have lots of original names that actually had meaning in our language. Now there are very few of them left, for example Vladimir (“one who owns the world”, masc.) and Snezhana (from “snow”, fem.). Most of modern names in Russia are of Greek, Latin, Scandinavian, etc. origin. So if your name isn't Russian there's still a chance to find its counterpart in Russian as well. Let's look at some examples.
Anastasia – Анастасия (Anastasiya)
Angela, Angeline, Angelica – Анжела, Анжелика (Anzhela, Anzhelika)
Ann, Anna, Hanna, Hannele – Анна (Anna)
Anya – Аня, pet form of Anna
Catherine – Екатерина (Yekaterina)
Darya, Daria, Tarja – Дарья (Dar'ya)
Elisabeth – Елизавета (Yelizaveta)