The phrase to keep a (or one's) weather eye open means to be watchful.
The phrase to make good (or bad) weather of it (of a ship)- behave well (or badly) in a storm.
The phrase to make heavy weather of means to exaggerate the difficulty or burden presented by (a problem, course of action, etc.).
The phrase to be under the weather - means to be indisposed or out of sorts
The phrase weather forecast means an analysis of the state of the weather with an assessment of likely developments over a certain time.