How to find which machine rendered a specific frame in a render farm

You can do this by adding custom metadata to the image using a Python script that is executed by each server. In the network folder, create a script named metadata.py with the following code:

import bpy, socket

name = socket.gethostname()

bpy.context.scene.render.use_stamp_note = True
bpy.context.scene.render.stamp_note_text = name

In your render command, add the -P flag with the path to the script, e.g.

blender -b path/to/file.blend -P path/to/metadata.py -o path/to/image.png -a

This stores the name of the machine in the "Note" section of Blender's metadata. You can view the metadata as described here to see which server is giving you trouble.

I don't know a very efficient way to do this, but in case you can't find a better solution you could have each machine render the same frame, then look which machine was unable to render it properly.