Placement of settlements is generally very far from random and those of the New World are no different. To use your examples, Santo Domingo was settled for better access to the ocean and for closer access to the gold mines in the south of the islands on orders of the Spanish Crown.^1 Jamestown was founded for rather different reasons. It was marshy swampland unsettled by the local Americans, reasonably defensible, and offered a natural port for boats. The NPS has an excellent page on the why's of Jamestown's placement that can be found here. I'll leave discussion of placement of the rest of the British and French settlements to other posters though.

Early Spanish settlement placement differed dramatically from the more organized systems enacted by the British and French. As noted by Stanislawski^2, early settlements were often far from resources and located in disease ridden areas. By 1513, the crown took note and gave general instructions for the placement and construction of new towns. They were to be located in clean areas, accessible by road or boat, have access to a source of clean water and arable land for farming. The crown's criteria for settlement locations continued to be refined through the next century. An additional requirement not codified by the crown, but generally practiced was that settlements would be near existing native settlements. This served two purposes. For the case of missions, access to the native populations for evangelical purposes was important. This was a significant factor in the settlement of San Luis Obispo in modern California. For other settlements, natives provided an important labor force, as well as access to resources and trade routes. The most famous example of this is Mexico city, built over the ruins of Tenochtitlan.

The access to resources is a huge driver in settlement placement even into modern times. This is especially true in the harsh deserts of Northern Mexico and the American Southwest. Arizpe and Hermosillo were both founded on relatively flat areas along the Rio Sonora. Phoenix lies at the confluence of the Verde, Agua Fria, Salt, and Gila rivers. Santa Fe and El Paso lie along the Rio Grande. San Francisco and San Diego both offer ports with excellent river drainage systems for agriculture.

If anyone were to pick a most important factor for settlement though, it would have to be water accessibility. There are no large cities in the Americas that do not have local access to water. This is most evident in the Sonoran, Chihuahuan, and Mojave deserts, notorious for their lack of perennial rivers. Outside the few perennial rivers, there are almost no large settlements. It is instead more common to find smaller population settlements that are able to store the temporary water from seasonal rainfall. This explanation offers another reason for the general trend of settlement near native population centers: Access to native water sources. In the case of Phoenix, the Hohokam canal system formed the basis for the modern Phoenix canal system. Similarly, Spanish missions with the Puebloans of the Colorado Plateau had a tendency to utilize the catchbasins of the natives for water during the dry season.

^1 Deagan, K.A. and Cruxent, J.M. (2002) Columbus's Outpost Among the Taínos: Spain and America at La Isabela, 1493-1498. Yale University Press

^2 Early Spanish Town Planning in the New World. Dan Stanislawski, Geographical Review, Vol. 37, No. 1 (Jan., 1947), pp. 94-105

British colonies in North America were not randomly assigned and were often thought out with the interests of the Crown. The Crown would give grants or patents for new settlements which served as the nucleus that would spread out. In many places natural boundaries were often used and these were normally major rivers that ocean going ships could navigate.

For example, the Massachusetts Colony got all the land north of the Charles River to the Merrimack River (that is almost the border between NH and MA now.) The Kennebec River in Maine was established as the most Northeast point the English would claim and beyond that was considered French Territory. In Virginia the James and other Rivers were considered as boundaries as the colony expanded from the coast. The Hudson was a big prize for the Dutch, who controlled it as way to trade with the different Indian nations. In the South, Charleston was founded on the Ashley River. Later the Georgia colony would develop along the mouth of the Savannah River

Grants and patents often allowed the settlers complete use of the river by pushing the boundary several leagues to the other side. This is the reason why Boston MA was allowed to develop and be part of Massachusetts Bay colony even though it on the other side of the Charles River boundary.

The importance of these rivers can not be denied as trade with the Indians (in furs especially) was needed immediately to make money for the settlers. Some important cities developed on these rivers. For example, Albany was chosen as it was the farthest point up the Hudson River that ocean going ships could travel. Similarly the cities of Springfield MA was the farthest that ships could go up the Connecticut River. In Maine, the city of Augusta was settled for the same reason.

In Virginia, the planters of influence always got the land on the rivers that ocean going ships could navigate. They set up docks to load and transport their goods to Europe as that was the location of the nearest market place.

So pretty much there had to be places for the settlers to live and trade with the Indians inland and also be able to ship items back to England. After all the settlements were meant to produce for the Crown.