Often when it's overcast I change my frame of mind to thinking of shots that will work as B&Ws. Overcast days are often ideal for B&W work because of the softer tonalities. The moodiness of overcast days usually mates well with B&Ws. For instance, check out a lot of Michael Kenna's work.
In he film days, it was cheaper and easier to process B&W. For the average hobbyist, it was difficult and expensive to use and process color film. Most digital camera can convert color shots into monochrome, sepia, etc.
In my recent trip to Fjorland, New Zealand, I made a mistake of switching to Monochrome because of the heavy clouds and fog covering the scenery. While the shots turn up well for B&W, I miss capturing the vibrant colors of the wildlife. Few days later in Blue Mountain, Australia, I stayed away from the monochrome setting and was rewarded in capturing subtle colors in the scenery that were faintly visible through the fog.
While I could possibly "FAKE" in the color by Photoshop, it is not the same, to me.