The digital video signal is the same for both DVI and HDMI hence there is no conversion of the video signal required.
HDMI 1.2 and later adds support for eight channels of digital audio. DVI and HDMI 1.0 and 1.1 support video only.
Did the ASUS monitor come with a .INF file? If it did you should manually install it.
Indeed, there's no advanced conversion needed. Just a cable with at one end a dvi and at the other hdmi.
What we need to know is whether he gets 1920x1080 at his desktop or not. If he does, then something very strange is happening here. If he doesn't, then I think it must have something to do with the graphic card drivers. Installing the newest one might fix it then.