Broadly speaking, dynastic names can be divided into three types – those based on places, those based on existing titles, and those chosen for their meaning.

As a rule of thumb, dynasties up to the Mongol Yuan opted for names that reflected their geographical origin or were the continuation of an existing personal title, while those from the Mongols onward chose names that had some sort of meaning. Going through the 'major' dynasties:

• The Qin 秦 was the continuation of the existing Kingdom of Qin;

• The Han 漢 was so named because its founder, Liu Bang, was given the fiefdom of Hanzhong (located on the Han River) after the fall of the Qin. (It was not, as a previous poster erroneously stated, based on the Warring States kingdom of Han, which was written 韓 and is pronounced differently in modern Chinese languages.)

• The Jin 晉 got its name from the Sima clan's control of the Jin River in Shaanxi.

• The Northern Wei 北魏 got the 'Wei' part from being located in the general region of previous Wei kingdoms – one during the Warring States and another during the Three Kingdoms/

• The Liang, 梁, Sui 隋 and Tang 唐 were so named because of their founders' having been the Dukes/Princes with those respective titles.

• The Liao 遼, established by the nomadic Khitan, were likely named after the Liao River in their homeland of Manchuria.

• The Song 宋 got their name from their founder's commanding his army of usurpation against the Zhao from the prefecture of Song.

• The Jin 金, led by the Jurchen, was named after the 'Golden River' in Manchuria.

• The Yuan 元, led by the Mongols, was a contraction of a particular term in the I Ching (Book of Changes): 乾元 – the beginning of the Way of Heaven and also signifying material abundance.

• The Ming 明 got their name from their origins in the White Lotus sect, which believed in a Manichaean prince of light who would drive out the darkness. The character Ming 明 means 'bright', hence the connection.

• The Shun 順, which attempted to supplant the Ming in the 1640s, likely chose the character for its meaning of either 'to obey' or 'to go smoothly'.

• The Qing 清 are quite hard to pin down, as unlike the Yuan they are not known to have issued a justificatory edict on their dynastic name. Two proposals exist: the first relates to the idea of Wuxing, the Five Elements. The Qing were preceded by the Later Jin 後金, which was the character for 'Gold' or 'Metal', while Ming 明 held connotations of fire (the character contains those for the sun 日 and the moon 月 as components), which in the Wuxing framework trumped metal because metal melts in heat. As such, the use of Qing 清 was intended to be auspicious as it strongly carried connotations of water, which would extinguish the Ming fire. This would make sense given Hong Taiji's more China-directed policy, so it seems to be the most commonly accepted etymology. Alternatively, Great Qing (daqing) 大清 was a transliteration from Daicing Gurun, lit. 'the Warrior State', but this is a much more tendentious reading. All the northern dynasties since the Liao and all dynasties since the Yuan had officially preceded their dynastic name with the the 大 character to form the Great Liao 大遼, Great Jin 大金, Graet Yuan 大元, Great Ming 大明 and Great Shun 大順, so it's unlikely the Qing chose Daicing first and produced Great Qing 大清. However, some suggest that there was a deliberate pun given the numerous characters with water connotations that could have been chosen instead.

This is a great question! That being said, I suspect it's more boring than you think. The most common name was simply the usage of a noble's former title as the title of the new dynasty.

Zhou Dynasty

The founder Wu of Zhou's family was first Counts of Zhou before ruling the kingdom.

Qin Dynasty:

This one is quite simple. The state rued by Ying Zheng (first emperor of the Qin) was called the state of Qin.

Han Dynasty:

Liu Bang established this dynasty, but the title he held prior was the king of Han (he ruled over the territory of Hanzhong) which lent its title to the name of the new dynasty.

Sui Dynasty:

Emperor Wen of Sui was the Duke of Sui prior to reunifying China.

Tang Dynasty:

Li Yuan was Duke of tang before he reunified China.

Song Dynasty:

This one is a bit different! Zhao Kuangyin didn't name it after his former title, but what he did name it after was where he was a military governor. So not too different.

I'm not too familiar with the Yuan/Ming/Qing dynastic names so somebody else will have to fill those in.

Here are some other dynasties as well:

Cao Wei:

Cao cao was the Duke, and then King of Wei prior to the end of the Han. Notably, Cao Wei was not the name of the state; it's an anachronism to distinguish it from the other Wei's in history.

Shu Han:

Similarly, Shu Han would have been called Han at the time of its existence. We add Shu because it's located where a state known as Shu once existed, and to distinguish it from every other Han you can think of in history. Liu Bei, founder of Shu Han, named it this because he didn't consider Shu han to be a new country, but merely the Han Dynasty continued.

Dong Wu:

Like the other two, there have been a lot of Wus in history haha. Sun Quan was king of Wu prior to being emperor of Wu, so that's where the name comes from.

Jin Dynasty:

Sima Yan was King of Jin before becoming emperor.

Chen Dynasty:

Even the only case of a dynasty being named with the family name of its ruler is only a coincidence. The Chen dynasty was founded by Chen Baxian, who was Prince of Chen.

Outside of the common method of taking a formal title and turning it into your dynastic title, the other common path seemed to be claiming an ancestral relation and using that as your state's name. Whether this was actually true or not depended on the specific dynasty in question.

One thing to note: it should be a bit obvious after the three kingdoms, but generally, when an extra character is added (especially if they're not directional) to a name, it's typically used to differentiate itself from another polity that existed in Chinese history. Examples: Shu han, Han Zhao, Bei Han, Nan Han, Hou Han.

However, sometimes, the additional character is added for other reasons. For example, the Zhou and Han dynasty are often divided between eastern and Western, while the Song is divided between Northern and Southern. This is not because they were separate states, but because there was a huge shift. For the Zhou, the switch from Western to Eastern was due to a switch in capitals and the collapse of royal authority (I'm simplifying here). The switch from Western to Eastern Han was due to primarily the xin dynasty overthrow/interregnum. The Song dynasty's switch from Northern to Southern was due to the Kingkang incident, where 2 emperors and much of the imperial court was captured by the Jin, leading to the Song losing control of Northern China and retreating to the South.