The best distro is the one that you are most comfortable with. This is because you'll want to keep it up to date with security patches and turn off services you aren't using.
Aside from that, I tend to prefer Slackware because, while it does have you editing configuration files by hand, it is very straightforward once you get past the initial learning curve (which is, admittedly, sizable). I wouldn't necessarily recommend it to you if you don't want to jump in with both feet. Then again, maybe the correct jumping approach is two-footed, if you're going to store bitcoins there.
Running bitcoind on Linux goes about like this:
You can fiddle bits and make it start automatically too, if you want, but I haven't bothered. I'm probably not the best person to help, but until someone else steps up, what do you have in mind for the website?