The Han Dynasty's most lasting contribution is most likely the POLITICAl system!
In the Han Dynasty, what excelled was the political system. The reason is because they had a less brutal government system. The government also managed to expand itself during the dynasty and even expanded china itself.
Following Politics there's Religion
Religion is huge part of China's culture, making this topic really important. The Chinese worship Buddha. It's like a god figure to them. They pray to this god just like how some people go to churches. Back then most citizens of china believed in Buddhism and worshipped it, but maybe it changed throughout time and not as much has this believe now.
Following Religion there is Economics
Money, trading, and business. Without these 3 a country or state or city even a village can't stay alive because of the lack of resources. Also because China is a large body of land and holds tons of people, trading helps feed them, without this system their business will run out and they won't have any money left. Therefore economics are needed.
Following Economics there's the social
Throughout the social system of china, we see people moving and settling, befriending, and trading. It's kind of like building relations for their own good. Social can be friends, family or just someone close to them. Without these relations, there would be no gain in the society that the Chinese lived in.
After that is the intellect of China