First, the keywords. My experience is with a Compaq Presario 1210 laptop (which is very similar to the 1600 series), and a pair of HP Pavilion xf235 laptops, but this should apply to most similar laptops; the advice is mostly generic.
The problem, in short, is that most laptops have their AC power jacks hard-mounted to the motherboard just like any other soldered-on component. But, unlike most other components, they regularly get yanked on when someone steps on the power cord or such, and eventually the solder joints (or the mounting tabs, which after all are pretty thin metal) start to fatigue, and eventually they crack. At this point, the laptop will only charge if the plug is twiddled "just so", and doesn't work reliably.
The official solution, of course, is to replace the motherboard -- which fixes the symptom, but not the overall design problem. On a laptop with a street value of $300, this is also not an economical solution. But it does mean that you don't really have anything to lose if you toast the motherboard trying to fix it yourself.
So. Enough preamble. Here's how I fixed the problem.
DISCLAIMER: THIS IS A DESCRIPTION OF HOW I DID THIS REPAIR. FOLLOW THESE INSTRUCTIONS AT YOUR OWN RISK, AND USE YOUR OWN JUDGEMENT; I MAKE NO WARRANTY THAT YOU WILL HAVE A WORKING LAPTOP WHEN YOU FINISH. SOME MINOR STEPS HAVE BEEN OMITTED.
On the Compaq, I used the solution that I found on http://www.delanoscientific.com/cpq1675.html. Essentially, you take a three-inch strand of some form of heavy-but-flexible two-conductor wire -- that site recommends a good grade of speaker wire, but I happened to have a dead iBook power supply that I salvaged some wire from -- and solder a socket on the end of it that matches the plug on the power cord, feed it through the original power jack hole in the case, and solder the other end to the motherboard where the original power jack used to be. One trick is to tie a knot in this new pigtail just inside the case, so that pulling on it only pulls on the knot, rather than pulling on the solder connections.
On the two HP Pavilion computers, I found a better solution. There is a hole in the case, right next to where the power jack goes, that's intended for a PS/2 keyboard or mouse plug. However, it's unused and blanked off, which makes it a great place to put a case-mounted power jack -- and it turns out that a standard 1/2"-diameter panel-mount jack fits perfectly. I used a Philmore No. 248 "DC Power Jack", which has 2.5mm x 5.5mm pin and socket diameters, thereby matching the original plug. While you're at the electronics shop, get a decent soldering iron and a power supply for it that lets you control the temperature. If you haven't done much soldering before, find some sort of discarded computer-board junk that you can practice on.
Here's what the new case-mounted jack looks like when it's installed; the original power jack hole is just to the left of it:
Actually doing the replacement took me about two hours, on the last one I did. The time-consuming part is that you need to completely disassemble the laptop to remove the motherboard. This starts (on the HP Pavilion computers) with gently popping out the panel above the keyboard that has the CD playback and volume controls on it, and continuing by removing every screw you come across. A digital camera is a very useful timesaver here; photograph the screwdriver pointing to every screw you remove, put the screw next to a number on a numbered piece of paper, and photograph it there -- then, go through the photos backwards and you have a detailed set of reassembly instructions.
When you have the motherboard all the way out, the next step is to remove the remnants of the old power jack, and solder on some one-inch-long wires (or the pigtail, if you're not using a case-mounted jack; remember to feed it through the case first!) in its place.
Here, it's worth noting that the original power jack has three pins, and acts somewhat like a switch. One pin goes to the ground, and becomes connected to the power-adaptor ground when the adaptor is plugged in to the laptop. The second pin is the +19.5V (or whatever the power voltage is) line; when the adaptor is not plugged in, it becomes connected to the ground pin. The third pin is a logic pin for determining when the adaptor is connected; it is open-circuit when the adaptor is not plugged in, and connected to the ground pin when it is.
This could theoretically be problematic with the pigtail-mounted socket, as that socket only has two connections -- the logic line for sensing whether the adaptor is connected does not exist. I simply ignored this problem on the Compaq laptop, and the computer seemed to work fine; it may or may not work properly on other computers. On the case-mounted jack, there are three lines, so this is not an issue.
Meanwhile, speaking of the case-mounted jack for the HP Pavilion laptops, there's the issue of attaching it in the case. Remove the blanking plate by appropriate means (I did this by grinding off the heat-swaged mounting pins), and install the jack in the hole that this leaves. You'll need to remove a bit of material from the inside top of the jack to prevent interference with the top half of the case later; now is a good time to do that. Also, if you want to be fancy, you can cut a small rectangle out of the blanking plate and superglue it in place to cover the old power-jack hole.
Anyhow, you then reinstall the motherboard (with its new wiring) in the case, and -- if using the case-mounted jack -- connect the wires to the jack. On the HP Pavilions, make sure the wires aren't over the LEDS immediately behind it on the motherboard; there are clear-plastic "light pipes" that have to go on top of those, and the wires can easily get in the way. Then reassamble the laptop, and that's "all there is to it".