This is caused by dodgy soldering on the connector which grips the screen flexicable. The pins at either end tend to make intermittent contact. To fix this you really ought to have a fine-tip temperature controlled soldering iron, a microscope, be used to soldering surface-mount components and have some experience of taking electronics to bits. You can probably get away without everything except the fine tip, but you'll need good eyesight! If you don't think you are up to it then I suggest you get linearlogix corp to fix it for you for a very reasonable USD 45. see: http://www.ipaqupgrade.com/. If your hardware is still within warrantly then you're probably best off just getting Compaq to fix it. They are reasonably quick and don't mind if it's been linuxed.
So, first take your ipaq to bits. The back is held on with 4 tiny torx-head screws. I didn't have a bit small enough but got them out with a normal jewellers screwdriver with only slight difficulty. The right tool would be much better. The case is opened by putting a fingernail between the case halves on the long sides and easing the back away. Once it is loose you should be able to lift it away (still connected by the battery flexicable) and then unplug the flexicable from the PCB (unusual connector). Remove the pen retainer bit of plastic (top left) sitting on the PCB as it'll fall out soon anyway.
To get the PCB out you need to unplug the backlight power cable (white+black). Remove the wires from the little slot they are routed in (noting how they should go back) to allow the unplugging. Also release the red and black wires enough to let you lift the PCB a bit (it won't come far due to the LCD connectors underneath). This lets you unscrew the 4 screws holding the rocker switch PCB in place. Remove this to release the rocker switch and thus the lower end of the PCB. The PCB is held snugly by the headphone socket at the top. Lift the bottom end so the power socket isn't jammed against the case and withdraw it backwards. A screwdriver between the headphone socket and case can help flex the case to release the socket. The main PCB is easier to get back in than get out.
Somewhere round this point the side switch bit of plastic will fall out. Now you can lift the PCB up enough to release the touchscreen (4 wire) and LCD (loads of wires) flexicables. The touchscreen is a conventional 'slide the ears back' connector (use a screwdriver to release it). The LCD has a rather funky connector which is released by flipping up the thin black plastic strip on the top. Take a look at how far in the flexistrip goes at the very edges of the connector before disassembly so you know when you've got it back in properly on re-assembly. Microscope useful here.
OK. So now you have freed the PCB and can put it on the bench for soldering. You can remove the LCD from the front panel by easing the red clip up one side and joggling things a bit. Then you can put the display connector back in (or have not taken it out yet!) and reconnect the battery to check that light pressure on the top of the PCB (display connector side) near the display connector does indeed make the screen flip back and forth - i.e. you have the same problem as me (and plenty of others with early ipaqs).
The dry solder joints are very difficult to spot visually (even with a microscope) - you really need to beep them out with a meter. Be gentle as you can bend the upper half of the connector pins (not fatal as the lower half is the contact but not a good idea). I found there were two or three dodgy ones at either end of the connector on the side nearest the top of the case. Soldering here is quite fiddly to get the solder to flow, and the LED gets in the way at one end. Persevere. You may need to resort to putting in lots of solder and then wicking it off again.
Now put the wide connector back in (fiddly). Make sure you put it back in as far as possible but be fairly gentle too - you don't want to tear or kink it. Make sure both ends are well-in and hold it in whilst pushing the black lock-strip back down. Check that the cable is in as far as it was before you took it out. (Mine was actually already slightly pulled-out at one end, probably due to stress whilst I worked out how to get the thing to bits).
Now it's just assembly all the way. Put the touchscreen flexcable back in and slide the clamp back in. Put the display back in the front panel if you took it out. Put the side switch bit of plastic back in. Put the rocker switch back in (make sure the springs are the right way up and the right side of the various bits of plastic, and the red and black wire is routed out the side). Put the 4 screws and LCD clamp bit of tin back in (make sure you don't stress the display flexcable too hard getting the last screw in). Put the PCB in place - headphone jack first then it should clip down, plugging into the rocker-switch board.
Plug the backlight power connector back in, and slot its wires into the little channel. Then make sure the black and red rocker switch wires go in the little slot in the same area and lies neatly along the board. Put the pen release bit of plastic back in place, plug the battery back in, clip the back in place and put the screws in.
Check it works.