The reasons for a particular city being named capital of a state, province, or country are as varied as the cities themselves. In that sense, this is a question of anecdotes rather than hard and fast rules. Many medieval rulers chose a capital based on the prestige associated with it (such as Mehmed II moving his capital to Constantinople after taking it, despite the dilapidated state of the city), while others focused on defensible sites. Similar reasoning went into the choice of Athens as the capital of the modern state of Greece (which was little more than a fishing village at the time).

Other times, the choosing of the capital was directly motivated by the fact that it was the site of the centre of power, and thus situating one's court and government there was a de facto necessity in order to exert the most influence over the country. Of course, often having your capital within the city itself was not necessarily desirable either; being close to the rabble, the disease, and the insecurity that both entailed for a ruling monarch often meant they moved their capitals elsewhere, even if the centre of power remained in that city. That said, this 'somewhere' else was often relatively close. In the case of Westminister for example, so close that said city is now part of London itself. On the other hand, there are cases like Versailles, which was considerably more distant, although Versailles itself is now part of metro Paris. Rarely, an entirely new capital was constructed, supplanting old cities entirely. The most famous case is, of course, Nova Roma, AKA Constantinople. In general, these cases are limited to larger powers whose governing apparatus is what drove the size and economy of the capital rather than a capital city whose economy was based around something else, as the former is relatively easy to re-direct, while the later is not so much.

Then we come to the New World, and the choice of capitals. Some are relatively straight forward. Mexico City, for example, sits atop the ruins of the Aztec capital. Others are not so much. Why was Washington D.C. chosen to be where it is for example? Well, aside from it's strategic location on an important waterway between two wealthy states, Washington itself sits almost directly halfway between the northern tip and southern tip of the United States as it existed when it was created. It thus was essentially an equal journey from almost everywhere in the young Republic. This sort of centralized locale is an important factor in the selection of many capitals in U.S. among states created since Independence; in fact, by far the majority of capitals are located very centrally within the state. Similar reasoning holds true for the Canadian provinces created since Confederation as well; a good example of this is in Alberta, where, despite Calgary being served by a more important railway with better access to the rest of the country, Edmonton was chosen for it's more central location.

Of course, there are other considerations too. Berlin for example is an interesting story as a capital. It was chosen because it was the capital of Prussia more than anything else, but it is arguable whether it was the best suited candidate in terms of prestige, size, economy, and even geographically. However, at the end of the day, given that it was the Prussian monarchy that created the new state, the capital was set there. On the other hand, when West Germany selected Bonn as the capital, it was mostly based on the fact that the city had offices still intact, a very pragmatic decision if I must say.

At any rate, I hope this helped give some perspective on the many reasons a capital could be chosen. Of course, it merely generalizes what is, indeed, a unique answer for every capital in the world.