mac mini video out

I have the same issue with my Mac Mini. Specifically the HDMI port. It takes a restart to get the screen to display. I plugged in an old cinema display via the thunderbolt port after my initial test and then the video kicked on. Prior to that the external monitor and a tv could not receive the video signal from the MM.

- Try resetting the SMC

Intel-based Macs: Resetting the System Management Controller (SMC)

- Try starting in Safe Mode if you can

OS X: What is Safe Boot, Safe Mode?

https://support.apple.com/en-us/HT201262

- Boot to Recovery and repair the startup disk

OS X: About OS X Recovery - Apple Support

https://support.apple.com/en-us/HT201314

- If it is repairable reinstall the OSX

How to reinstall OS X on your Mac - Apple Support

https://support.apple.com/en-us/HT204904

- If you do not have a backup use disk utility to restore the internal disk to an external disk so you can try to recover data.

Then format the startup disk and do a fresh install of OSX