Chinese dynastic titles usually had something to do with titles or territories their founders held before becoming Emperor. Those titles themselves were often related to vassal kingdoms during the Spring and Autumn period. Of course, this is not a universal rule, but it applies to most dynasties. For example, Li Yuan was the Duke of Tang under the Sui and was in charge of the city of Taiyuan, where the Spring and Autumn state of Tang was located. For this reason, he named his dynasty Tang. Zhao Kuangyin was military governor of Songzhou during the Later Zhou, where the Spring and Autumn state of Song was located, so he named his dynasty Song.

Sometimes, people claim descent from emperors of previous dynasties and name their states after that. Liu Yuan, a Sinicized Xiongnu, claimed Han ancestry and so he named his state Han. The Later Tang state was founded by Sinicized Shatuo Turks who had been awarded the imperial surname of "Li", so they base their claim on the fact that they were successors to the Tang.

But before the barbarian conquest dynasties, all dynastic names were taken from Spring and Autumn kingdom names. The Khitans broke this tradition by naming their dynasty Liao, which was purely based on geography. The Jurchens similarly chose Jin because according to one theory they rose from near the Jin River. Kublai chose Yuan because of its Confucian connotation, as he wanted to attract the support of the Chinese. Zhu Yuanzhang chose Ming probably because of his roots in the messianic White Lotus/Red Turban movement. The Manchus chose Qing because the character symbolized water, which could put out fires (which symbolized the Ming).

These dynasty names are usually based off clan names (though not exclusively). Han is associated with the surname liu (刘).

The Han dynasty (206 BC - 220 AD) and the warring states Han were definitely related to each other. The founder of the Han dynasty was Liu bang, who was directly descended from the warring states Han family.

Southern Han (917 - 971) just happened to take that name because the emperor happened to have the surname liu, and he chose to rename his Kingdom to match it. It had no strong relation to the earlier Han dynasties.