First, set Port Range Forwarding on your router for port 3306 to your server. Then, create a MySQL account for your developer so they can log into MySQL on your server. If you know the IP they are coming from and their IP doesn't change, you can add more security by creating the MySQL account to only work from a specified host (their IP address). You will also need to make sure you have MySQL setup to allow remote access. If you are on Windows, you just need to go through the MySQL setup program again and check the box near the end of the setup process that let you allow remote connections. If you are using Linux, then it is a bit more complex than that, but the Debian Linux Setup Wiki page describes how to do it. Then, they should just need to setup Navicat or any other MySQL tool and they will be able to do edits.
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