Final answer:
Sumerian civilization, as an early great Mesopotamian civilization, began in the early 4th millennium BCE, around 4000 BCE, with significant developments occurring around 3000 BCE.
Step-by-step explanation:
The Mesopotamian civilizations, particularly the Sumerians, began to take shape in the early 4th millennium BCE. By 3000 BCE, the Sumerians had established city-states that served as religious and administrative centers and were beginning to enjoy economic prosperity due to the advancements in agriculture, trade, and technology. This timeline is consistent with the emergence of city-states and the inception of Sumer as one of the first great civilizations. Therefore, the correct answer to the student's question regarding when Sumer, one of the earliest great Mesopotamian civilizations, began is C) 4000 B.C.E. However, the specific context of the question which includes the description of the Sumerians' political organization and economic prosperity most closely aligns with the developments around 3000 B.C.E.