I think it depends on what kind of name you want. Henry is very classic. While not as timeless as something like Edward or William which haven't dropped out of the top 100 since 1904, I know several Henries my age, older than me and younger than me. Some people might find a name like Henry dull, but it's solid, classless and is unlikely to drop hugely out of favour.
Felix is trendier. It has the 'x' which is a letter of the moment in naming at the moment. It's shaken the cat association and is growing steadily from outside the top 200 to into the top 100 and still rising. I would predict it following Hugo into the top 50, and maybe further into the top 25. I think Felix will date more than Henry, but it's a more interesting name. I also love the meaning of Felix if that's important to you.