Any of the options are fine--it really boils down to your personal taste, so, of course, none of us can answer it for you, but we can give our opinions, like you asked. Personally, I would not use Jedd, just because I don't see the point of adding a letter. Jediah sounds a little strange to me, but maybe only because I'm used to hearing Jedidiah. I love the whole long thing overall, but I do think the two Ds so close sound a little funny. I would possibly change the name a little to a mix of Jedidiah and Jeremiah and make it Jederiah.

What do you think of something like that? I just noticed my typo in the previous sentence--you might also like the spelling of Jedidiah, and he could possibly even go by Jedi.

Well, I'm getting carried away! I would go ahead and put the longer name on the birth certificate since you like the full name and the nickname. If you didn't like the longer version, I would just use Jed. (Our son is named Luke. He is not Lucas, just Luke. Luke is about the only version of the name we like.) I hope I helped!