The top half of my right shifter fell off on a trail the other day, so I need to replace it, but I'm not sure what my options are. The bike is a 1997 model Trek 6500 ZX with an 8-gear rear cassette and Shimano Deore derailleur. I can't tell what model the existing shifters are - they don't have gear indicators and the only identifying marks are the word "Shimano" and "8s".
Any 8-speed Shimano mountain bike shifter will work. All Shimano 8-speed shifters use the same cable pull.
Addressing what model your actual shifters are, a `97 8 speed MTB with a 'Deore' drivetrain probably has Deore XT M739 and Deore LX M569 parts. (See the Shimano Wikipedia page which includes a timeline of MTB groupsets.)
You could narrow down the exact model you have, then try to find a lightly used or old stock replacement on EBay. Googling for 'M739 Shifter' for instance yields images that you can compare to what's on your bike.
Many shifters from about this time were combined shifters and brake levers. If you have cantilever brakes be aware that the brake levers for cantis and linear pull (v-brakes) are different. Newer shift-brake combos will be for linear pull brakes and will not work with cantilevers.
I don't think Shimano currently makes an 8-speed groupset above Tourney level. But you could also look at newer 8-speed shifters from the Alivio/Acera goupsets.
I got an 8 speed no-name shifter off Aliexpress to convert a bike from 9 to 8 speed.
It's made of a metal like substance and works fine with shimano dérailleur for $10