yes. boxes are custom built for specific sub parameters allowing for port tune, and proper mated frequency response. You'd likely end up with a worse sounding product if you randomly selected a budget sub driver and stuck it in your existing box.
If the box is sealed, your idea has merit, because sealed boxes are more forgiving than ported for mismatched driver parameters. If it's ported, put it by the curb, or let somebody on craigslist think they found a good deal on a box.
16 years of age means you got your money worth.
If you really want to play download winisd and hit the DIY forums and see if the guys can help you match a driver to that box.
Problem is you'll end up paying more for a plate amp and 10" driver for an underperforming product than you can buy in something like the aforementioned Klipsch RW-12D on newegg for $300 right now.