As you write it, be sure you have the names and dates right. If you have a question, check back with the person who told you about it. If you only have a nickname, find out the full name of the person, and make sure it's spelled right. Use the same technique for place names. It will save you a great deal of grief later on.
Pictures are important , too.
If you have a family gathering, take photographs to remember it by. Then write down the date and place of the photograph, and the names of the people in the picture. Store the details with the photographs so they can be true family heirlooms, rather than just pretty pictures for future generations to wonder at.