A correction is in order. Namely, the Mongol Yuan was not the first state based in China to opt for a symbolic rather than a regional name: it was in fact preceded in doing so by the Jurchen Jin, whom the Mongols conquered before the Song, and indeed the Yuan was declared before the Song were defeated. Why the Jurchen Jin opted for that title is not wholly clear, given that the Khitan state they had conquered did have a geographic title as it was called the Liao in reference to the Liao river in southern Manchuria. Whatever the case – we simply don't know – this may be what gave precedent to the use of a symbolic title on the part of the Mongols who conquered them, and their later Manchu descendants who established the Qing. This is necessarily somewhat speculative as none of these states, as far as I am aware, gave an explicit reason for why they chose not to use a geographic title, although the Yuan did explain their choice to use yuan in particular (it derives ultimately from the Book of Changes) even if they didn't explain why they were using a symbolic title in general. This is more than we have got for the others.

The Ming most likely named themselves as such in reference to the idea of their being a force of light to drive back the darkness which they associated with Mongol rule, as they were spearheaded by Buddhist-Manichaean syncretic sects whose cosmology involved these sorts of dualistic struggles between light and dark. This may also serve to explain the lack of a geographic name for the new state: it was not founded by regional nobles whose power was originally rooted in that region, but by a relatively grassroots religious movement with no firm geographic centre. Again though, there is no explicit information so we can only give speculative answers.

The Qing, as ever, are more complicated because there has been some effort to identify some kind of reasoning behind the term. At present, there are three possible explanations, which, it should be noted, can all be true at once:

1. The Chinese character qing 清, meaning 'pure', contains the water radical (氵); whereas the character ming 明, meaning 'bright', contains the characters for the sun (日) and the moon (月), and thus has associations with fire. While we can't say for sure why qing in particular was chosen out of all the characters that might have a water component, we can say that there may have been a cosmological reason involved, and that they chose a name that implied extinguishing the fires of the Ming. There is one problem with this, that being that it presumes the Qing state was, in its conception, an anti-Ming endeavour as opposed to perhaps a more regionally-focussed power, not helped by the fact that much of the source material that has traditionally been used comes from after the Qing conquest of China, when said conquest was naturally easily construed as premeditated.

2. The Mongolian transliteration of da qing 大清 is daiiching, which might have been a pun on the Mongolian word daiichin, 'warrior'. There is some degree of contextual support for a Mongolian-based interpretation, as the renaming of the state from (Latter) Jin to Great Qing came after the Qing seized the old Yuan imperial seal from Lighdan Khan. The problem, of course, is the question of how far Mongolian may have been influential.

3. Some Japanese scholarship from the turn of the millennium has suggested that early Manchu documents sometimes use the phrase amba cin gurun interchangeably with daicing gurun, suggesting that there could originally have been an intended near-homophone in Manchu such that the Qing were both the Great Qing in Mandarin and the Great Cin in Manchu, but by the mid-late seventeenth century the simple transliteration of daicing came to predominate in Manchu usage. Cin is defined by Jerry Norman as '1. chief, principal, main; 2. straight, straightforward; 3. the south side; 4. a small white heron'; we can probably discount definitions 3 and 4 here, and suggest that, if it was indeed the case that cin was an intentional and meaningful Manchu nomenclature, that the Qing had a Manchu name that meant something like 'The Great Primary State' or 'The Great Steady State' or something along those lines. Whether we can assert that qing derived from cin, however, is very much unclear and there is as much case if not more for it being the other way round.

And of course there's the quite plausible scenario that all three are true: the Manchu leadership hit on a term that had one meaning in Manchu or Chinese, and a usefully meaningful near-homophone in the other, and also happened, when the Chinese was transliterated to Mongolian, to have some pun potential there as well.