I have a relatively well-known ham-related personal website. I am currently modernising its design and would like to promote ham.stackexchange.com.
What is available to do so? Can I gain some reputation by doing so?
(This answer addresses only the “gain” part of the question.)
You cannot gain reputation by promoting the site. However, if other people visit a question using the special URL you can get from the “share” button, then you can be awarded the Announcer, Booster, or Publicist badges.