I agree that site has a bit of a throw in the deep end.
I had a player or two that did the same thing (postively moronic move to have solder as a mechanical joint if you ask me) but that is surface mount stuff.
Technically it is correct but it is aimed at far smaller components that what you are likely playing with which changes things a bit and is about adding stuff for the first time where there is a piece of metal to bond to (solder does not bond to polyboard all that well).
One of two things is likely to happen:
1) you lift a layer from the polyboard
2) you lift the trace
If you do not break the trace try and rebond it. Superglue should be fine for polyboard (assuming it is not a multilayered one) but for other stuff it is best to use some resin (hot superglue evaporates and it stings like hell when it gets in your eyes, not to mention it sticks to everything (see a dodgy forensics show: it is a fingerprint grabbing tool)).
If you do break the trace then coolbho3000 is on the right track, I usually try to create a new solder pad as soldering to a trace especially for a mechanical type joint is no fun (unless you are in schoolboy PCB realms in which case it is usually OK). Jumping it to a unused section is a good one (be damn sure it is unused however) or attaching a piece of foil.
Another thing I like to do is make a tearaway strip with foil, thing about tearing a sheet of paper almost in half so you have two "legs", holding it by the "feet" you still have a little bit of give before it goes unlike a solid joint which will cause what you see when you started the thread. Sidenote I read something this in a robotics journal a few years back and while it is fairly obvious to me (how many people make the point of failure in something easy to replace) it may be covered by a patent or three somewhere.