I've searched the internet far and wide and can't figure out why my 360 Elite won't display 1080p over HDMI.

It's going to a Westinghouse LVM37w3se over the Microsoft supplied HDMI cable and the 1080p option is grayed out.

The only fix I've found involved an HDMI switcher, something I don't have. How do I fix this?

Why is this even a problem? It allows 1080p over component, but not HDMI!

Infuriating.

The problem is your TV didn't report correctly (via EDID info) to XBox that it supports 1080p. Component is a one way communication that XBox outputs regardless your TV accept it or not.

If your TV doesn't support 1080p/60 over HDMI (my Pioneer 940 Elite does not - it only accepts 1080p/24 via hdmi) then the 1080p option will be grayed out in the Xbox setup. Yes it's annoying. :(

Are you going to the HDMI port or one of the DVI ports?
You could try an HDMI-to-DVI cable to go through the 1080p DVI port.
Some of the ports on Westy monitors are configured for TV use (with overscan and 1080i input) and others for PC/Console use (1080p with no overscan). You just need to pick the best port for the application. Not sure what the config looks like for the W3SE, I'm only familiar with the older W3 which had 1080p on one DVI and the HDMI, and the second DVI was 1080i.
You can always use the VGA port, too.

Edit: Hmm, it looks like it should work.
You *Are* going direct to the display, right? Not going through a receiver?
I don't think a bad cable could cause this, so you may have a sick Westy... :-(
Connectors

QTY TYPE FORMATS SUPPORTED
1 HDMI(R)-HDCP + L/R Audio 480i/p, 720p, 1080i, 1080p
1 DVI -HDCP + L/R Audio 480i/p, 720p, 1080i, 1080p
1 VGA/D-Sub/RGB + Audio mini jack 480i/p, 720p, 1080i, 1080p
2 YPbPr Component Video+L/R Audio 480i/p, 720p, 1080i
1 S-Video+ L/R Audio 480i
1 Composite Video + L/R Audio 480i

another vote for faulty EDID here, i'm 100% sure thats your problem. seems like nearly half of the SE models have this problem.