I like the name
Eugene much more than just
Gene. For one thing,
Eugene keeps the meaning of the name intact. Secondly, as other posters have mentioned,
Gene can sound like
Jeanne. If
Jean was a female (great grandmother?) maybe you can save that for a girl's middle name some day.
I think
Henry Eugene sounds good.
Best of luck!
(My name list:
http://www.babynames.com/namelist/9516801)