What condition results when the cuticle becomes overly dry and splits?

The condition where the cuticle becomes overly dry and splits is known as a hangnail. A hangnail occurs when a small piece of skin at the base of the nail pulls away and can become inflamed or painful. This can be caused by a lack of moisture, leading to dryness and cracking. Proper hydration and care can often prevent hangnails, emphasizing the importance of keeping the cuticle area moisturized.

The other conditions mentioned, while affecting the nails or surrounding skin, are distinct from hangnails. Nail separation refers to a condition where the nail plate detaches from the nail bed, often unrelated to the cuticle's condition. Onycholysis is the medical term for the separation of the nail from its underlying tissue, which is typically a result of trauma or certain health conditions. Paronychia is an infection of the skin around the nail, often caused by bacteria or fungi, which is also not directly related to dryness or splitting of the cuticle. Thus, hangnail is the correct answer in this context, as it specifically describes the issue of dryness and splitting at the cuticle area.