I don't know your situation entirely, but guessing from the fact that this is an older system, I would bet that there are a lot of other problems with it, and MySQL's PASSWORD() function is probably the least of them. In fact, you could probably store plaintext passwords and wouldn't lose much (though since encrypted passwords are already there, you might as well keep them). This isn't to say it should never change, but you could probably spend time on other parts of the system first.

Again, this is just from someone completely unfamiler with your situation, so take all of the above with the properly-sized chunk of salt.