My old Westell router is so out of date that it does not have WPA-2 capability, nor can it communicate via 802.11g or 802.11n standards. HTC-One seems to only work with g or n unless you use the "HTC One" trick (which doesn't work when there are 2 devices with the same name on the same Wi-Fi network).
Update Feb 28, 2014:
I use Verizon for my home phone + DSL. I have had DSL service for almost 10 years. So, I called them to ask their advice on how best to handle the issue with the old DSL router and its lack of communication with my HTC One smart-phones. Verizon agreed to send me a new DSL router. Thank you, Verizon!
My Wi-Fi network is now happily working with all old and new devices
Holy Cow, I stumbled on this forum because I have the same problem where my HTC One won't connect to my home wi-fi. The phone connects to every other wi-fi I try, and every other device I have can connect to my home wi-fi, it's just this one phone that won't connect to the one wi-fi. I tried the device renaming trick, and to my surprise it worked! I tried renaming it to something else, and it didn't work at first. Then I renamed the phone to "HTC ONE", and that did the trick!