How I got HPE iLO 3 working on a modern version of Firefox.

This is done by going into your iLO Settings, forcing AES/3DES Encryption, then changing a value in Firefox config, which is what I will go through in detail below.

Step by Step Guide.

Step 1: Enter your iLO Web Interface from a browser which supports it. (To do this, I would recommend getting a portable, old version of Firefox from https://ftp.mozilla.org/pub/firefox/releases/, 10.0.12ESR from 2015 seemed to work for me.

Step 2: Navigate to Administration > Security > Encryption. In here, you need to edit the value for 'Enforce AES/3DES Encryption' and change it to True.

Step 3: In your main Firefox Browser, navigate to 'about:config' and search for 'security.tls.version.enable-deprecated', and set that to true. (Enabling this means that you are able to access sites with versions of TLS which have been depreciated, so be careful.)

Conclusion

Once you have completed the above steps, you should be able to access the iLO 3 Web Interface from a modern browser, obviously you will still have the Self-Signed Cert error, however you can bypass that.