I switched over to GitLab for a static blog when I realized I could use the free shared runners to compile any static site generator out there. Such a beautiful feature. I would be interested to hear how it holds up in a production environment.
However, most of us prefer to generate such things from templates and markup files. The most popular system for doing this, by far, is Jekyll (https://jekyllrb.com/), probably because Github recommends it. Metalsmith (http://www.metalsmith.io/) and Wintersmith (http://wintersmith.io/), while not as popular, are certainly options if you're a JS fan, and don't mind cutting code to write your configurations, and have some very nice features for customization. However, if you're coming from Wordpress, I'd reccomend Lektor (https://www.getlektor.com/). It's Python based, well documented, and uses a simple file tree to build the website like all other options discussed. However, it also provides a simple admin panel, which can be used to write your posts, manage your posts, and publish the generated site to your server.
As for distributing a static site, you just have to stick it onto some kind of hosting. S3, Apache, Github pages, whatever. Once again, Lektor can do this via the admin panel. For the rest of the generators, you're on your own: you'll have to write the scripts yourself, but this is fairly trivial.
Danger recently moved their site over (https://gitlab.com/danger-systems/danger.systems, https://danger.systems/) if you want another example.
We have a bunch of working example sites in a bunch of different static site generators at the "pages" group: https://gitlab.com/groups/pages
Jekyll, Middleman, Pelican, Metalsmith, Octopress, Nanoc, etc. etc. All working examples. We also have gitlab-ci.yml templates if you create the YML file via the web editor, so for many of these you can get a simple site deploying pretty quick.
And if you want a bit more guidance we have a series of blog posts on static sites & GitLab Pages: https://about.gitlab.com/2016/06/03/ssg-overview-gitlab-page...
Personally I'd recommend Jekyll, but it really depends on what languages you're comfortable in :)
(Shout-out to Marcia, Axil, Kamilt, and others for helping build out Pages and these examples over the last while, really awesome resources!)
Not much to see, but it was pretty straightforward to make, you just write the templates.
The resulting .html is pushed with the source to the github repo. This setup, while not very sexy, doesn't even need a CI.