Two of the most popular choices will be Ubuntu, or a RedHat based system (usually CentOS). Here's my experience of running live servers on Ubuntu and CentOS over the last 5 years:
Ubuntu is 'easier' to get started. It tends to use newer packages and there's more to install out of the box. This is good if you want to set something up quickly and easily. It's easier to upgrade Ubuntu from one version to another (for example, 10.04 to 12.04), but obviously you'd need to check the applications you're running. More customisations and custom applications = harder upgrade.
Ubuntu has a large user base and an active community, to if you want to set something up there's usually "walkthroughs" that will tell you every command you need to enter. Typing random commands into your server because someone tells you do can be a terrible idea, but it's a useful thing to have if you're less experienced at running servers. Just stick to reputable sites for walkthroughs.
CentOS is a good choice for something reliable. It tends to include packages that are quite new but have proven themselves reliable, rather than the bleeding edge. It also has fewer packages installable by default, but you can always compile software yourself, enable another package repository, or find an RPM online (I won't do this to my live servers). It comes slightly more locked-down out of the box, but this can be changed. Upgrading from one version to another (i.e. CentOS 5 to CentOS 6) isn't easy, so you're probably looking at setting up a new server and moving stuff across. However, they do support their releases for a ridiculously long time. My live Moodle servers are based on CentOS 6, and that's officially supported until 2020. By that time, we'll probably have moved to different hardware anyway, so having to upgrade the OS isn't a big deal.
I have four live servers and five development servers running CentOS, and one live server and one development server running Ubuntu.
One last thing: if you want to use BigBlueButton, that'll need to go on an Ubuntu server. There are instructions for setting it up on Ubuntu, but trying to get BigBlueButton running on CentOS is a bag of hurt.