Despite phones with oversized screens, YouTube still doesn't allow for HD playback on Android. Thankfully, modder theos0o has created an alternate version of YouTube that enables HD playback, even over 3G.
There are multiple versions of the APK, one set for Froyo and Gingerbread users, and one for Ice Cream Sandwich users. You can also choose whether you want the Wi-Fi version (which gives you the option for HD over Wi-Fi) or the Wi-Fi/3G version (which gives you the HD option on both Wi-Fi and data).
Once you've downloaded the APK, you should just be able to install it like any other sideloaded app: transfer it to your device, run it from your file manager of choice, and off you go. The modder recommends deleting the
/system/app/YouTube.apk that's already on your device (which you'll need to be rooted to do), but none of us at Lifehacker had this APK on our system -- all we needed to do was install it and start watching videos.
The installation process and forum comments lead us to believe that this is still a very "iffy" mod, so your mileage may vary. Our intern couldn't get it working on his device, but I got it running swimmingly with almost no effort. As soon as I started watching videos I got an "HD" button in the upper right hand corner that produced noticeably better video. So, to try it out for yourself, hit the link below to download the mod -- though we recommend you root and make a backup first, since you never know what could go wrong.