WordPress allows you two great choices:
1) One site where each person has their own "category," such as Bob, Jane, etc. They each also have their own login, and they might be able to set their own default category for posting. If not, then they can also use the dropdown menu to select the category. With the categories option, a sidebar on your blog will have a category menu so if you want to see all of Bob's posts, click on Bob and you'll get them. Go to the main site and you'll see everybody's posts again. Your family can use "tags" to further differentiate post types, such as "vacation, work, etc." Or you can continue to add categories so they can double categorize it, but that will clutter up your Categories menu.
2) WordPress can be installed as "Multi User," so each family member has their own separate blog, such as bob.example.com, jane.example.com, etc. This will isolate each family member to their own website, so this might not be desirable in your situation.
You can create a section of Profiles as a group of Pages. The main Page, which is reachable from the top menu bar in many cases (or in the sidebar), can have each member's name and picture if desired. Clicking on that person brings up a profile page, and you can add a link to their category so clicking on the "Bob's Posts" link will bring up all of the blog entries by Bob.