Is it possible for a discolored beauty mark to be skin cancer?

Yes. Change in the color, size, margins, bleeding, nodule formation may indicate transformation of a mole/beauty mark to melanoma. The patient is urged to see her/his doctor for examination and if warranted, removal of the lesion to be examined by a pathologist.
Possible. Remember your abcd's when thinking of melanoma. Look for moles with asymmetry boarders that are irregular color differences within the same mole diameter greater than 6mm also any new mole or old that starts to itch, bleed or ulcerate should be of concern. Another rule is the ugly duckling rule. Which refers to a mole that looks distantly different from the others on your body.
Yes. Skin cancer can arise from any part of the skin, no matter how normal it has looked for years before. This is one of the things that makes skin cancer so challenging. See your surgical oncologist or dermatologist for a complete skin examination to see if you need a biopsy or not.