Before I start, a short disclaimer:
I am not responsible for anything in case this goes wrong. Do it at your own risk, and use some common sense.
I've tried this on various Ipaq devices and it worked pretty well, but I have no idea if/how well it works on other devices.

What is needed:
1. A PocketPc PDA with a broken screen.
2. Optional: some screwdrivers to open it.
3. The software Pocket Controller (30 USD, but has a free 3 days free trial).
4. A PC with Windows (should work with 98, 2K, XP and more)

A quick search on eBay will reveal many PocketPC enabled devices with various flaws, most common being a broken screen.
Their price is pretty good, usually about one third or even less than a fully functional device. However, make sure that the PDA comes with the following things included: Battery and docking cradle. A power adapter would be nice as well, although you can usually buy them separately from eBay. If one of those items is missing, do a search and see how much would they cost to buy, and see if the whole deal (broken PDA+ missing items) sounds good to you.
One other hint is to actually contact the seller before you buy it, and ask if the PDA does turn on despite the broken screen. If it doesn't, don't buy it.

The good news is that there is a way to use those devices, for many things such as:
1. Playing internet radio stations.
2. Using it as a portable phone with Skype.
3. If it has a camera, you can use it in conjunction with your WiFi connection to monitor various areas in/around your house.
4. As a 100% working emulator for software development.
5. All kind of servers.
6. Possibly other uses such remote control of a device via IR and various other things that require some electronics experience.

The first step is getting such a device. The best source is eBay, you can find good deals just about anytime.
Next, after you have the device, analyze the screen, and see if the digitizer is broken. You can check that by taping the screen and seeing if it makes any noise.

Step 2 [optional]. If it is broken, you might consider opening the PDA and removing the screen altogether, it's going to save some battery power and about 20 grams. An advantage to doing this is being able to see inside the device, which some people find to be cool. Nevertheless, this is an optional step and removing the screen might cause further damage if you are not careful or do not have the right tools. Speaking of the right tools, you will most likely need some special screwdrivers. I recommend getting them from a store hardware store such as Sears, rather than buying them online. The advantage of buying them from a store is that you can take the PDA with you and see which screwdriver fits.

Software installation:
Download and install Microsoft ActiveSync (free, might even come with the PDA if a CD is included).

The next thing is downloading and installing the software Pocket Controller from www.soti.net . It is not free, but has a 3 days free trial. I guess that you could always write your own software to emulate it, but I personally think that it's well worth 30 USD.

Now it's time to connect your PDA to your PC. Make sure the battery is charged, connect the cradle to the PC, then place the PDA on the cradle, and turn it on.

ActiveSync will pop up and say that a new device was detected. Follow the instructions and set a name for it and do whatever else ActiveSync asks you to do. At the end of this, the ActiveSync icon in the system tray will be green.

Start Pocket Controller, go to Tools, "Install Device Software". It will say something about looking on the PDA's screen to see if there are additional steps required. Ignore this.

Go to File, Connect, select ActiveSync(USB) (or COM, whatever you are using).

Now, most likely what you will see on the screen is some welcome message asking you to calibrate the device. Oops, that program actually wants input from the hardware digitizer (touch screen) and you can't do it with the mouse on the Pocket Controller screen.
No problem, just go to Tools, Explore. Once you are there, there will be a device "Mobile Device". Click on it, then go to the folder: Windows/StartUp.
You will find a file called welcome.lnk or something similar. Delete it.
Again, go to Tools, Soft Reset Device.
The Pocket Controller connection will be lost. Wait for a while, then reconnect again.
Now you should see the standard PocketPC screen, and you can touch it with your mouse.

You can do a lot of stuff here, just use your imagination. Google for PocketPC software and you'll find a lot of free and not so free programs you can use.
Most of the PocketPC 2003 devices already come with Windows Mediaplayer 9 so you can use it to listen to mp3 files and various radio stations.
Skype has a PocketPC version which works very nicely, but it requires a relatively fast CPU (400 MHz or more) and a WiFi capable device. Fortunately many devices do have such a CPU, and built in WiFi is pretty standard now.
I use such a PDA with Skype and it's really cool to be able to go around the house talking on the phone for free (or for little money).

Good luck, and if you have questions or comments please post them here.

Radu Privantu