This node-gyp permission denied issue happens when, as the root user, you globally install node modules that require compilation with node-gyp. As root is the default user on Digital Ocean this seems to affect many people and should probably be fixed at the root cause.
Luckily there is a pretty simple workaround for this:
npm config set user 0
The other solution is to create a regular user and install everything as them. This is safer as installing modules as root means they could do anything to your system. I’d recommend installing all node modules as a normal user without root access if you’re on a machine with anything important on it.