There are many reasons for this important incident which came as an order from Allah to His Messenger (وأنذر عشيرتك الأقربين) (And warn your near relatives). Allah Knows all the reasons but many of the Ulama think the reasons can include; 1. His family members know him and his sincerity and truthfulness more than others.
2. Main reason according to many great Ulama is to prove to his family that Ali is his main supporter and his successor.
Wassalam.