I'm not crazy about Barney as a stand-alone; it kind of sounds like an adjective. Barnaby is nice, but I like Barnabas better (but I have a thing for -s names like Elias, Matthias, Tobias, Linus, Silas, etc.) All of the Barn- names remind me, positively, of The Dark is Rising (Barney Drew was officially Barnabas).
Jackson and Jenson are both surnames, which doesn't mean you can't use them as first names if you love them, but Jackson has been popular as a first name for a much longer time. Also, think of how the names would likely be shortened - Jack is classic, Jens is less familiar for English speakers, and Jen feels like a shortening of Jennie, Jennifer, etc.
I think Barney, Jackson (those two are close), Jenson would be my order of preference.