A few things I'd like to note down.
1. 'Chat' not easily found on the site. I agree with a few comments that it's not prominent enough on the site. You have to delve your way through the menu to find it, and even then the majority of people will be unaware of what it's purpose is. I disagree with that it should be added as a main link on the bar, it would make the bar far too cluttered for my liking, especially when new sections of the site are released over time.
How to get around that? On the home page there's the box that welcomes people to the site, advertises the forums and also the Smiler pages - why not add chat to that too? There are only three frames on there now, I'm sure a fourth one would not hurt (although arguably chat may not be that pivotal to warrant being advertised there).
Make use of social media! One thing that this site aims to do that no others quite so is making an emphasis on the 'community' - using utilities such as Facebook and Twitter could draw in more of an audience that may get more involved.
A bar could be added to the top of the page during chat nights too (much like the chat bars that the team on here used to do on TTF) - it's a quick form of advertising it and can pull in a good load of members.
Of course the forums is a good way to advertise chat too. Ben already does that with The Tavern - however may be global announcements could be made instead? If you're a guest, or someone who rarely looks in the Tavern that's already a fair few people that could have been drawn into chat.
Obviously a few other things need to be addressed before working out when to advertise chat and so on, but I'll cover a few things in a moment.
2. Official chat nights. Most obvious suggestion. I'd suggest there's a weekly 'official' chat, chosen on a day where the site is busier than usual (all sites have there busy and quiet days) and a team member is able to join chat to ramp up excitement. It's simple enough to do and could link up with advertising it weekly. Over time I think more traffic would go towards chat and eventually it would become a standard things that all members do weekly to join together.
3. Chat games. May involve a bit of work on the teams side (or by other members - look below). Create simple games. A pub quiz - all it would involve is a form being made by someone, even if it's made by members and used with Google Docs, just something simple! Picture games. With radio you could even have a 'guess this sound' game.
4. 'Chat team' - I know this will come across as a bizarre idea, however I thought I might as well throw the idea out. As we know the team have a busy job for running the site and forums. So an alternative to keep chat popular could be to get, say, two or three people to form as a part of a 'chat team' - they could be responsible for advertising chat, drawing people in, monitoring people in chat (I know modding isn't a big thing in there, but it's still nice to have a bigger voice in there) and creating games.
It would give the team more time to do their work on the site but allow chat to have some momentum. The chat team (in my eyes) would be something that is kept separate from the main team however, they would not come under 'TS Team' (in the bigger picture) but would form as a new small team that works differently.
5. Another suggestion that may be a bit weird, why not just have a member or anyone that's willing to - to run off a small radio channel. There's a fair few free platforms out there. They don't have the necessarily talk, just have some music playing. So people in chat can just have a little music to listen to while in there and even suggest music to be played. Obviously this is not really a favourable suggestion, but again, just throwing what I can think of out there.
I will not lie either, a monthly Joely Radio would be lovelyyy.