This is quite difficult to answer without more information in regards to what's not working specifically.
Although there are three probable causes:
* ***The display drivers in the device are corrupt and need a reinstall.*** Considering it's a Chromebook, I'd rather just go with reinstalling the entire machine.
* ***The HDMI port is damaged.*** If this is the case it can be replaced, but might not be worth it.
* ***A motherboard issue.*** In this case, you're better off to just buy a new computer if this is a major issue for you. If the issue lies within the motherboard, it would require microsoldering knowledge and a lot of work to fix, and would definitely cost more than purchasing a new computer.
Have you tried using another cable or/and monitor? The problem might not necessarily lie within the computer.