Shahzedah, is a Persian term which clearly, as you mentioned means the Son of the Shah but more generally the meaning is a term for any Prince of the family. The term was used widely within cultures influenced by the Persian courtly culture such as the Mughal Empire, Timurids, Ottomans, Safavids, and others.

Sources:

• The Venture of Islam Volume 2 & 3 by Marshall G.S. Hodgson
• The Muslim Empires of the Ottomans, Safavids, and Mughals by Stephen F. Dale
• Islamic Gunpowder Empires: Ottomans, Safavids, and Mughals by Douglas E. Streusand