I think it would be OK to use a Russian name, but it might be easier to choose one which has a more "English" nickname option, which they could use. For example,
Aleksei could go by
Alex. Otherwise I think the best ones would be ones which are Russian but relatively well known in Elglish-speaking countries too:
Anya,
Larisa,
Tamara,
Anton,
Leonid,
Roman.
I would defintely avoid using names which are particularly hard to spell and/or pronounce correctly in English, such as
Zinoviy,
Kseniya,
Fyodor,
Evgeniy.