The term "Manifest Destiny" was coined by John O'Sullivan in 1845 during the lead up to the Mexican-American War and the annexation of Texas. It was a term describing an already commonly held belief that Americans were destined to settle the west. Here is a link to the article:

http://www.americanyawp.com/reader/manifest-destiny/john-osullivan-declares-americas-manifest-destiny-1845/

Trying to nail down when the idea came into being though is difficult. At its core, Manifest Destiny is just the idea that the lands to the west belong to Americans by right. Really at any point in American history this can be seen as true.

An argument could probably be made that the original settlers in 1607 and 1620 had some semblance of this idea, especially the Puritans that wanted to establish a new England to glorify God. However, I don't know if that is necessarily the best way to view Manifest Destiny's origins as a popular idea, though it's definitely the foundation.

Even before the American Revolution, there were settlers pushing further and further west. The French and Indian War was fought largely over encroachment in the Ohio River Valley. It was a pain in the ass to deal with for the British but apparently a popular enough course of action to start killing each other over it.

Even shortly after the Revolution, ideas of entitlement about westward expansion were prevalent. The Northwest Territory (Ohio, Indiana, Illinois, Michigan, and Wisconsin) and the Northwest Ordinance really set into motion the idea of Manifest Destiny as we know it. The ordinance established rules for continuing to gain westward territory and forming that territory into states. There was no major public push back against settlers moving west.

I would say where the idea really became popular and what it is today is in the early nineteenth century with the purchase of the Louisiana Territory. It nearly doubled the size of the country, and it was acquired in a way that was much less tense than the circumstances surrounding the Northwest Territory. All of that land to the west belonged to America and no one else. It was their God given right to move into the area and begin working the land.

Really, all of American history has had this entitlement to lands to the west. Sometimes it was a pain in the neck for the government, other times not as much. Either way, the government rarely ever made people stop pushing west.