If you want to get rid of a family member quickly, and permanently, you can choose to throw them into the Volcano.
 
Open the State Map, locate the volcano and zoom into the County Map view. At the top of the Volcano you will find a shrine you can tap on and select which Noble(s) you want to send to a fiery death.
 
If you are not in a rush, and want to help out another family, you can ask anyone if they want your unwanted Noble and have them send a proposal. Accepting a proposal adds your Noble to their family.
 
You can increase the size of your family by researching ''Family Size'' through the Academy. This will let you have more family members in your Keep before you have to make room for more.
 
Sacrificing a Noble while assigned to an Army will cause any attached troops to be destroyed as well. The army must be disbanded first to prevent accidentally sacrificing your troops.