Final answer:
Confucianism(option A) is the tradition that became fully developed during the Han dynasty, serving as an ideological framework influencing governance and culture.
Step-by-step explanation:
The tradition that has its roots in the early dynasties but only became fully developed during the Han dynasty is Confucianism (A). Confucianism, a system of philosophical and ethical teachings founded by Confucius, played a significant role in the ideological and administrative development of China. While it took on a distinct form during the Han period, Confucian thought was present in various capacities before this era, influencing governance, culture, and society. However, during the Han dynasty, it became more structured, serving as an ideology that justified power and influenced imperial dignity and the administrative practices of the state.