The iPhone 5 is pretty good, and for most folks, its Wi-Fi is actually very good. It was in our tests. But a lot of people have been reporting in that the Wi-Fi is actually awful, and can't hold a signal at all. This should fix that right up for you.
First off, this isn't the same issue as Wi-Fi data bleeding onto your carrier plan (though the two could certainly be related). This new complaint has been observed by people holding an iPhone 5 in one hand, and an iPhone 4S in another, and the 4S getting full Wi-Fi bars while the iPhone 5 struggles to keep even just one for a few seconds.
Head over to the iPhone 5 discussions on Apple's boards, and you'll see that this is a pretty common problem. But what has worked for some people is switching their router over from WPA/WPA2 over to WEP.
WPA and WPA2 are newer encryption technologies than WEP, but they have a few more issues syncing on networks with older devices. If it's an issue, the newer WPA-enabled devices are supposed to be able to automatically switch back and use WEP. It's possible the iPhone 5 is having problems with that. You should be warned, though, that WEP is significantly less secure. (Update: Just to be totally clear, this should only be used as a solution temporarily, until Apple fixes this through a firmware update, or another solution is found. Until then, though, know you're trading security from people trying to access your network for functional Wi-Fi on your phone.)
The switch to your router is relatively simple. You'll need to access your Router gateway, which should be easily found through Google. For example, most Linksys routers are 192.168.1.1, while Western Digital routers can be accessed by simply typing "wprouter" into your URL bar.
From there, just go to the Wireless and Security options, select WEP instead of WPA, and enter your new password. All done! Hopefully your Wi-Fi issues will be cleared up. Let us know if it works—or doesn't—in the discussion below.