Personally, I love drawing large, stylized eyes. Doing so can bring additional emphasis to that part of the character or composition. It can also change how your work communicates. Generally speaking, cartoon eyes tend to take advantage of exaggeration for expressive or aesthetic reasons.
Here are some basic examples of how the eye could be exaggerated to create emphasis or to further craft a stylized aesthetic. I tried elongating the eyes, enlarging the eyes, and shrinking/minimizing them.