I have gone through a bunch of meta SE sites to see if they have any site promotion ideas that we hadn't though of yet. Here is what I found:
- A leaflet or flyer, see e.g. the leaflet for English Language & Usage. (This idea was also suggested on Russian Language SE.
- A subreddit, as proposed on Linguistics SE.
- A "Language learning flashmob week", like the stats flashmob week proposed on Cross Validated.
- "Polishing" the questions with the highest number of views, as proposed on French Language SE.
- A question asking competition or question challenge week, as proposed on Chinese Language SE (although there was no response when the idea was repeated two years later). The idea came from a StackOverflow blog post from 2012: Hot Topics: A Contest Formula that Works.
- Contacting language-learning communities such as Polyglot Club, as suggested on Russian Language SE.
- Writing a great blogpost, as suggested on Chess SE. (This requires publication on a blog with a significant amount of traffic.)
- Creating our own blog, like Worldbuilding SE's blog on Medium, as suggested by @fi12 in our meta question about YouTube videos. SE announced in December 2014 that it would no longer start community blogs for individual sites. But it turns out that @PythonMaster already created an unofficial LL SE blog.
- Creating a video. See also Making a promotional video for a Stack Exchange site? and Featuring Stack Exchange on YouTube - Anything I Should be Aware of? on Meta SE.
- Retweeting upvoted questions on Twitter, as suggested on Chess SE. That is ... if LL SE has a Twitter account.
- Promoting the site on LinkedIn, as suggested on Cognitive Science (although it seems that they never did this).
- Find closed questions on other sites that may be relevant here and migrate them, as suggested on Open Date SE. This raises the question how to find close questions on other sites. See Where can I find all the closed questions? on Meta Stack Exchange.
- A somewhat similar post on Philosophy SE.
- In addition on how to promote the site, someone on Cognitive Sciences SE asked, what is the best time of day to promote the site?.
- Publishing guest posts on blogs about language learning (since there are so many language learning blogs), as proposed by Hatchet.
We can't do all of these things—let alone simultaneously—but we can do some of them, one after another.
Update: I put a slightly modified copy of the above text on Meta Stack Exchange, where it is more likely to be found by participants of other beta sites. (The changes concern references to LL SE itself.)