I bought this to play over LAN with a friend but cannot due to the GOG installer using a generic key. I don't see a serial key anywhere on the game menu in my account like some other games do. Is there any way to change the CD key after installing?

ArmA 2 came with a GOG.com serial key change utility. But I don't see one for ArmA 1...
The Arma 1 serial is a value with the name "KEY" in this registry entry:
HKEY_LOCAL_MACHINE\SOFTWARE\Bohemia Interactive Studio\ArmA
Sure you can back it up and remove it and restore it. But I don't know if it will trigger a prompt for a new serial.

Thanks. I'll fiddle with it and post how it goes.

Edit: Deleting the key only generates an error about a bad serial. It doesn't prompt you to input another serial key. The serial goes through some kind of conversion for the registry key.

Edit Again: Someone on Github posted a solution to convert keys to the hex format for the registry key entry.

The last post by liskin (Jul. 6th, 2019):
https://github.com/ValveSoftware/Proton/issues/767
Terminal Command, replacing 1234-etc with a real key:
"echo 1234-56789-ABCDE-FGHIJ-KLMNO | perl -ne 's/-//g; tr/IO/10/; for $i (0..2) { $res = 0; for $j (0..7) { $res += index("0123456789ABCDEFGHJKLMNPRSTVWXYZ", substr($_, $i * 8 + $j, 1)) << (5 * $j); }; printf("%010x", $res); }; print("\n");'"

GOG should still provide unique keys and let you input at install or offer a key changer but this is a workaround in the meantime.

Nice find!

It may work only for Steam keys, not BIS original ones, generated before 2010 year. But it is not accurate)))

Update: yes, I checked and found that Arma-1 keys are stored in registry according the same rules as for Arma-2 etc.