Apple - Repurposing "menu" button on Windows keyboards used in OS X

It has neither purpose nor name in OS X.

Maybe Karabiner could do it, if you can figure out what key code it sends - try Key Codes to see if it generates anything usable.

To expand on @Tetsujin's answer: you can indeed achieve this with Karabiner. Here are some detailed instructions.

In short: go to the Misc & Unistall tab, open private xml, change it to:

<?xml version="1.0"?>
<root>
  <item>
    <name>Menu key to option right</name>
    <identifier>private.menu_key_to_OPTION_R</identifier>
    <autogen>__KeyToKey__ KeyCode::PC_APPLICATION, KeyCode::OPTION_R</autogen>
  </item>
</root>

Go back to the Change Key tab, click Reload XML, and tick the new mapping you just made at the top of the list.

Karabiner has its own keyboard EventViewer in the Misc tab, so you don't need KeyCode.

Doing @Steph's answer but with Karabiner:

1. Install Karabiner from here
2. In "Simple Modifications" tab, select your keyboard (mine was this "Microsoft.." one)
3. Press on "Add item" to make those 4 modifications in the screenshot below

This will map your keys to be as follows: