I learned this trick reading Backpacker Magazine years ago while waiting for an air taxi flight into the Nahanni River. It really works.
Make a cup of coffee. I like to add cream and sugar since it makes the bubbles easier to see but black coffee is fine. Hot tea or hot cocoa will work too.
Step 1: Watch the Bubbles
When you pour the coffee into your cup, watch the bubbles.
If the bubbles move to the edge of the cup rather quickly, that's a good sign. Expect clear skies for the next 12 hours.
If the bubbles hang around in the center of the cup, get out your rain gear. You can expect rain in 12 hours.
If the bubbles slowly move to the edge of the cup, you may get a bit of weather, but it should be clearing in a few hours.
If you've managed to make a cup without bubbles, flop a spoonful of coffee back into your cup and make some more bubbles.
Step 2: Theory
The theory behind this trick is that high pressure will push the bubbles to the edge, and high pressure indicates a period of sunny, calm weather. Low pressure won't move the bubbles and low pressure systems typically bring unsettled weather.