Some icons are born as icons (talking about semiotics) cause they was created to be similar to the object/concept rapresented. But, as time passes, some icons are become symbols (talking about semiotics) because they are used for convention and not anymore for analogy.
The are many other examples:
When it comes to popping up a contextual menu, the overflow icon is now widely used for that purpose. If the majority of websites your users visit already use this convention, It makes sense to leverage on this learned mental model already.
So to answer your question: use an overflow icon (horizontal or vertical is fine). You may even add a border to it on ...
Aline has a great link to research in his answer. I'll add that if the icon is on the left, you leave the right side available to add a downward pointing arrow to indicate a dropdown menu. Additionally, after some digging, I found way more examples of icon on the left than on the right:
According to the article Where to Place Icons Next to Button Labels, from UX movement, the icon should come first:
Eyetracking research has shown that users scan from the left in a vertical movement. In order for icons to serve as a visual scanning aid, users need to see them before they see the button label. Placing them to the left of your button label ...