For those who don't know, here is what a LAN party is. It consists of having a group of gamers bring multiple gaming platforms (computers, Playstations, etc.) to one place, like a home, and connecting them together either by ethernet cables or over an ad-hoc wireless network. The point is to have gamers be together to be able to watch one another play and trade appropriate trash talking in person. It usually also consists of having party style food and music as well, typically lasting deep into the night and often turns into a more macho "sleepover".
It almost seems too obvious to have to explain how this could be used as a discipleship or evangelistic tool, but here it goes. LAN parties have a number of things that lend them to being effective ministry tools: they're safe (in the sense of not being too "churchy" to possibly scare away some folks), they're personal, and they instantly make a connection between everyone who attends. Those three elements ensure that LAN parties can be effectively used tor each out to lost gamers in a community by those in the church.
The really great thing about LAN parties and one of the things that sets them apart from other methods of gamer ministry is that you don't even have to be a gamer to effectively host one. Most LAN parties are BYOS (Bring Your Own System) anyways so it doesn't require even owning a gaming device. Since junk food is a necessary component of every LAN party (cut to every LAN veteran nodding in approval), the host can make a connection simply by being gracious enough to open their home and provide some Mountain Dew and assorted junk food. It can even be educational to those who are unfamiliar with gaming culture and is a great way for non-gamers to work with gamers to reach out to the gaming community. A non-gamer host and a gamer planner would be a dynamite partnership for an outreach centered LAN party.
Like with the previous method, LAN parties also require a level of sacrifice. Opening one's home is almost never easy for any home owner to do, and for a Christian home owner in particular to be opening their home to those who may have a very different lifestyle than they do is certainly never easy. It also requires effort to provide refreshments, possibly equipment, and space for a successful LAN party. It may even require being okay with video games that perhaps push some uncomfortable buttons. Call of Duty and Halo are both shooters and LAN party staples. Both of which are games many church members I've met have issues with.
However, to church members willing to make those sacrifices, it is perhaps one of the simplest and most effective ways to be able to make personal connections with gamers in your local communities and to be able to show them the Love of Christ.