I like the way 'Yevgeniy' is pronounced in Russian - sounded simply beautiful and exotic. Personally, this name gives me some vibes that its bearers may have kind of reserved character.
A gorgeous name, but for me, I'm only familiar with it through the name of the villain of the third Metal Gear Solid game. So, if your kid hangs out with a bunch of video game enthusiasts, it may be a little odd, but other than that it's much more pleasant and masculine sounding than its english equivalent, Eugene.
As a Yevgeniy myself, I am Russian. However, I don't believe in the American translation of this name to Eugene. I personally go by Justin. Other Russian people I know go by simply Yev. Still others take names like Gary. Some Russian females with this name go by Jen or Jenny. I personally would go by Yevgeniy if I did not have to constantly tell people how to pronounce my name. If you're Yevgeniy considering an American name, you could go by Jerry. I do not recommend anyone to change their legal name because I believe you should be proud of your roots.