You are talking about just one family, even though it is made up of more than one person, so the correct form would be: "My family has got brown eyes." It would be plural if you said it this way: "The people in my family have got brown eyes." In this case it is plural because 'people' is the subject, not family. 'In my family' just modifies 'people'.

Normally, we use the singular. In some cases you can use the plural. Technically, when talking about a group you use the singular. When talking about the individuals in a group use the plural. That is, each person in the family has his or her own brown eyes, so you can say "have". They do not share all of their brown eyes. However, brown eyes is a trait of your family as opposed to blue eyes in another family. Therefore, "has" is also correct. To make it more clear, lets say "one eye" instead of "brown eyes". My family has one eye. --All members of my family share a single eye. Incorrect. My family have one eye. --Correct. All members of my family only have one eye each. (Note that you could say "In total, my family has one eye" meaning that only one person has one eye, and everyone else in the family has none.)

In my family, we all have brown eyes. Probably the most natural way to talk about tendencies for families to share physical characteristics is to say: [XYZ] RUNS IN MY FAMILY / TENDS TO RUN IN MY FAMILY ... Brown eyes tend to run in our family. Mathematical ability tends to run in our family. Having a hot temper tends to run in my family. I've read that breast cancer is hereditary i.e. it runs in families.