HDMI is a relatively high-bandwidth video standard, meaning it requires more than one Cat5 cable to send a full 1080p signal... but you can still send HDMI for relatively long distances over Cat5! Here are a few adapters you could use to do it:
VGA might be the older standard for computer and HD video display, but it works with practically any device—no matter how new or old—and is the standard for things like projection, PC inputs on HDTVs, etc. You can send VGA (and SVGA, XVGA, etc. resolutions) over Cat5 for very long distances with little or no signal quality loss, if you use an extender like the one below:
There's a great writeup over on Life is a Prayer.com concerning the best way to get an analog Composite signal (RBG) over Cat5 cable.
Standard video signals (analog) can be sent over the twisted pairs of a Cat5 cable for very long distances, and if you use Cat5e or shielded twisted pair cabling, you can go even further! In many churches and auditoriums, our cable of choice for video distribution is always Cat5. Run a wire to each TV, and then distribute a video signal over the Cat5 network using a set of 'baluns' (which stands for 'balanced / unbalanced' - they make an analog a/v signal run long distances over data cable.
Here are some of the baluns we recommend:
We'll post more ways you can use Cat5 cable for non-network data traffic soon.