As a reader, I don't find anything about the author's name important.
However, female writers often use their initials if writing from a male perspective, or in a male-oriented genre like thriller or mystery.
J.J. Jones vs. James J. Jones vs. J. Jeremiah Jones vs. James Jeremiah Jones
As for your examples, "James J. Jones" would be appropriate if there's already another "James Jones" out there. James Jeremiah Jones seems lengthy, but no doubt still workable, and "J. Jeremiah Jones" I find to be a little strange, since it masks the first name but not the middle. Again it would be something to think about, though, it there was already another "James Jones."
Sometimes the genre you're writing can play into the byline you'd like to have.