First off, I wish you luck on this. I really really hope that this kind of idea takes off. Arcades have been fighting a losing battle for a while, and a new outlook is needed to rebuild that community feel.
There was this fancy "upper class" pool hall in my city close to where I live. It went under years and years ago, but the building hadn't been leased to anything else since. I was always daydreaming of taking it, and turning it into an establishment much like you mentioned... So since I don't think I'll ever pursue it, I'll pass my ideas on to you.

The "Public Lan Party" idea is sound, but from what I read earlier, you need to definately take into consideration that if the costs are more than around $10 per person for an entire night, you probably won't get many groups walking in. Most people will save the extra bucks, and go to a buddies place with their PCs, and spend the money on caffeine and pizza (which is cheaper when you eat at home).
To try to break out of that, I was thinking of creating an atmosphere that wasn't strictly networked machines. A few ideas were:
1. Tournmanets. Yes, many many tournaments. One weekend would be devoted to a multi-tier Starcraft tournament, the next weekend could be Counter Strike: Source, etc. People would pay for entry to the trouney ($5 or so) and you could offer up a decent videocard or something equivalent as the top prize. Runners-up would get coupons for future activities at your location, perhaps.
2. Projectors. Multi-purpose. Not only would they serve as excellent screens for displaying the action from the tournaments, but they could also be used for movie-nights. For example, on your week's designated "slow-night", you turn off the PCs and instead host an Anime night. Every week could consist of a different collection of films for people to enjoy. They pay a small cover at the door, and voila, you've just reached out to a new audience. Projectors can also host sporting events (Superbowl, etc) which you could advertise and bring in even more.
3. Liquor License: Granted you won't be able to serve alcohol without closing your doors to minors, but on specified nights (or simply "every night after X:00" the place serves alcohol. It would be perfect for the gamers who have now hit their mid-20s to 30s. These are also the gamers who would probably take up many of your extremely late hours. (I drink beer and stay up until at least 5am at the LAN parties I attend)
4. Advertise advertise advertise. I can't stress how many neat little places in my town went under simply because nobody knew about it. Word of mouth will only get you so far, especially when it's such a niche market to begin with. So flyers, ads in the paper, local commercials, etc. You can never overdo it.
5. I don't believe this would apply to everyone, but the place I was daydreaming of had a really good layout for a stage. Which means that on slow-nights, I could also bring in a local band or two to generate some revenue. Probably not recommended, though.
6. Sponsor a game of Assassin in your town.

If you don't know what that is, I'll be brief... Everyone pulls a name out of a hat, you study the nae you're given, and hunt down that player. You only have basic information to go on in the beginning, and it's a matter of tracking them down. If you do, and spray them with water, they "die" and you get the info of the person that they are hunting. Wash, rinse, repeat until there is only one.

The only reason why I suggest this is because, well, everyone I knew as a teenager WANTED to do this, but never got off his butt to organize it. That being said, it's great advertising.
Some of the suggestions are a little out-there, I know, but you just never know.
- FA