[internet-explorer-8] What do I need to do to get Internet Explorer 8 to accept a self signed certificate?

You can use CertMgr to add a certificate as a trusted publisher or if it is self-signed, as a root certificate

CertMgr.exe /add CertificateFileName.cer /s /r localMachine root

See Microsoft's documentation here:

https://docs.microsoft.com/en-us/windows-hardware/drivers/install/using-certmgr-to-install-test-certificates-on-a-test-computer