Please enumerate what tests you have tried. If you haven't, test a direct connection to a HDMI device (TV or monitor) with a good quality HDMI cable. I consider this almost a rule: if a cable came with a monitor, it is crap. Using dongles and conversion cables like you have described can be an issue themselves.
Yes tested direct connection with several HDMI cables that are good. It was working fine. It stopped working. The monitor is fine. I hooked my laptop to test it so that is not the issue. I own 4 NUCs never had any HDMI issues.
Ok. Wish I could say that this is the first time that I have heard of the HDMI port failing, but it is not.
After verifying your warranty is still active (here: http://www.intel.com/content/www/us/en/support/services/000006303.html?wapkw=warranty+check), you need to directly contact Intel Customer Support to initiate the RMA process (you cannot do this through the forums). Here are pages where you can lookup contact information, including local phone numbers, sorted by geography:
You can use the online forms service or email, but I usually recommend calling them directly. The above links provide local or toll-free numbers in most countries and support most local languages. You could also use the chat service (start here: http://intelsupportchat.force.com/icslivechat/ics_tech_desktop_ww_english_Chat), but understand that this service is offered in English only and is subject to the normal (9:00-5:00 M-F) office hours in the Pacific (UTC-8) time zone.
Hope this helps,