I would say the easiest Linux to use if you want people to seemlessly use it between their home computer (Windows XP) then you would want to use Linspire or Freespire.
Most 2.6.x builds of Linux support WiFi adapters, however it does always depend on the adapter and the manufacturers' support for the *nix OS.
Also FreeBSD is Unix not Linux, quite a bit of difference and "Alpha" is for Sparc Alpha processor systems, you would want i386/i686/x86 variants for an Intel/AMD-based system.
To be perfectly honest Windows 2003 or Vista Business are probably the best options for Games, User-Friendliness and Administration.
They give you much quicker, and better access over what the users are doing, what they're allowed to do, log-in instances, etc.
For these OS, they will also both seemlessly boot as client only machines from a single machine even over a WiFi. Although if you do use internal WiFi remember that you will have a max of 54Mbps over (upto 8/16 connections) as opposed to a normal wired router that will provide a minimum of 10Mbps (generally 100Mbps) per connection. You also have less chance of external hacking from other WiFi connections, less need for setting and maintaining security keys as well as potencial connection issues with the WiFi adapters dropping connection due to positions of furniture or people.
Windows generally speaking is also far better for connection specialist hardware; for example an internet café they had in LA used a card system. So people would register and top up using their credit cards. That way they can use a machine whenever it's free and the user-time ran until you ran out of credit so you're not being over/under-charged.
As Kentaree has noted, remember there are more costs than the initial costs. Quite a few costs you wouldn't realise prior to opening because of not thinking it through properly.
Something that would bring people in would be getting a license and a few Xbox 360 consoles. Lan parties are big things, particularly for games like Gears of War or Halo. It's good business for younger users having somewhere they can either play the latest games or play games online when they don't have an internet connection at home.
