Replace mystery front derailleur
If you are not sure what you need, take it to the bike shop. Front derailleurs are complicated! You need the derailleur that fits your bike design and the gears that are installed on that bike. 

There are five things to look at: attaching it to the bike, how many chainrings it needs to handle, the space it has to maneuver, whether the front shifter is indexed (sometimes it was only the rear) and how the FD is connected to the shifter.

The attachment for MTBs is just 2 ways, an E-type (which is the fairly rare bottom bracket mount) or a band clamp around the seat tube. You need the right band size (the clamp) for your seat tube - old steel frame MTBs usually had 28.6, or if it's aluminum, it's usually a 34.9. This number is the diameter of the seat tube, in millimeters - you need to measure.

Most of the older MTBs were built with triple chainrings, so most MTB front derailleurs will support that (you don't want a narrow cage double or one destined for the road, though it would probably fit the space).

If the shifter is not indexed you will have a much easier time, if it is indexed, then you will need to be more particular in picking a FD - one that matches your front shifter.

Now it gets complicated...there are top swing and bottom swing FDs. You need to be able to clamp the derailleur to the seat tube at the right height above the chainrings, but it can't be so low that it hits the chainstay when it swings to move the chain. This is the complicated part. Depending on the frame, and the size of your chainrings, either could fit. Older MTBs usually had bottom swing FDs. But there were interfering bottle bosses and weird high chainstays...who knows.

You have already said that the cable goes under the BB shell, so you will need a bottom pull FD. If it went under the top tube, then you would need a top pull FD. Shimano make a dual pull, so it could do both (very handy for cyclocross bikes!), but if you had one, you still need to choose how it swings.

Given the info you have provided, I would expect you to look for a Shimano 28.6, bottom swing, bottom pull...something like the FD-M650, which is a 90s era Deore DX. But I could easily be wrong :)

Good luck!