If you have a spare computer, a linux solution will be cheaper. check out
nocat.net, it'll perform authentication for you, redirecting all web requests to a login page till they authenticate, then they can go where they want. You didn't mention whether you wanted to restrict access to your network or whether you just want it open to anyone in range. I'm assuming you want to restrict to paying customers.
nocat doesn't need a super powerful computer, any old pentium or pentium II will be fine. So you should be able to set up authentication for cheap. For a router/nat/firewall checkout one something like smoothwall or ipcop.
Note that you'll have to take some time to read and understand how to set this stuff up, but it'll save you money. Judging from the price point you set $100 - $200, I'm assuming you want things on the real cheap. Of course if you want a fully boxed/ready to run system they are available too as Goosemaster pointed out, but that unit is more than twice what you wanted to spend.