This thread discusses the mechanism for choosing related questions at a high level. I'm wondering how those questions are then displayed on the RHS. Just going through a few examples by hand, it's definitely not by vote score, and it doesn't seem to be in order of relatedness. The API mentions a "rank", but every example I've looked up has had the related questions ordered simply by question ID (ascending). Can anyone confirm or deny this?
As Joel explains here, Elasticsearch is being used to give the results.
So the order is most likely by Relevance, same way search results are ordered.
When searching for something, the default sort is "relevance":
Each search result then have a relevance score, generated by Elasticsearch. It's hidden in the markup, for example the first result in above search has score of 205.0617 and the second has a score of 93.1004, etc.
In the related questions list I couldn't find the score, but pretty sure the sort is by that score behind the scenes.