Your name might come from the Slavic name of Danube, Dunaj, as
Nanaea noted, or it might be derived from the old Russian "dunai" - breath. In favor of the "breath" theory - Ukraine is nowhere near the Danube, but close enough to Russia (an in fact, a former Soviet Republic) to have absorbed Russian words. In favor of the Danube theory: Ukrainian last names generally end in -enko. -evic names are typical for the better part of former Yugoslavia: Serbia,
Monte Negro, Bosnia and Herzegovina, etc. So, your family might indeed have come from that region, in which case the name can very well mean "of the Danube"