This is an excellent question, that sounds suspiciously like an essay question I answered during my BA. Regardless.

The answer is somewhat complicated, people didn't call it the Renaissance for quite some time, Renaissance is a French word, not Italian, it was probably first used to describe the period in the 19th century, by a French Historian.

Leonardo Bruni, the Florentine Humanist, Historian and politician was probably the first to divide history into three periods: Antiquity, Medieval and modern (or now) he even dated it. Antiquity ended with the deposition of the last Western Emperor Augustulus, by Odoacer, in A.D. 476, the medieval period ran between then, and sometime in the 11th or 12th centuries - a period marked by German invasions with weak claims to the imperial glory of ancient Rome. The modern period began with the triumph of the Guelphs over the Ghibbellines, (factions supporting the pope and the Holy Roman Empire respectively) and the demise of the Holy Roman Empire as a force in Italian politics. Bruni's ideas were adopted by his successors, Poggio Bracciolini, Nicolo Machiavelli, and we still largely think of History in these terms today, albeit with slightly different dating.

The point however, was that Bruni was the first to express the idea of living in a modern age. If you read some of the works of Florentine chroniclers it is clear that they understood the significance of the period that they lived in, and recognised it as being different to what had preceded it.

Matteo Palmieri wrote a manuscript in about 1429, On Civil Life, in a passage entitled The revival of the arts he wrote this:

For some centuries now the noble arts, which were well understood and practised by our ancient forebears, have been so deficient that it is shameful how little they have produced... before Giotto, painting was dead... Having been restored by him, sustained by his disciples and passed on... painting has now become a most worthy art... Sculpture and architecture, which for a long time had been producing stupid monstrosities, have in our time revived and returned to the light, purified and perfected by many masters. As for literature and liberal studies... those who should have been the leaders and true masters of all the arts have been neglected... Latin written on paper or carved on marble is a rough caricature of what it should be. But today we see our Leonardo [Bruni] of Arezzo sent into the world as the father and ornament of letters, the resplendent light of Latin elegance, to restore the sweetness of the Latin language to mankind. For this reason anyone of intelligence should thank God for being born in these times, in which we enjoy a more splendid flowering of the arts than at any other time in the last thousand years.

The Italian word Matteo uses often in this passage is: rilevare translated as 'revived', and 'restored', modern translations would read it as, 'noticed' or perhaps 'discovered'. The Renaissance was a rediscovery of the genius of antiquity. Matteo repeatedly identifies art forms which had become corrupted or gross in the preceding centuries, and points to their recent revival and mastery by contemporary figures.

Giorgio Vasari in his The Lives of Artists (1550) was probably the first to use the term La Rinascita literally: rebirth. He saw the gothic art that followed the Roman Empire as a decline, that was only arrested by the rising to prominence of artists like Giotto and Cimabue. Jules Michelet, a French Historian probably defined the Renaissance first in his History of France (1855), but was mostly concerned with making the Renaissance a major part of French history. Jacob Burckhardt, a Swiss Historian was the first to postulate a theory of a Renaissance that was uniquely Italian, he wrote his seminal work The Civilization of the Renaissance in Italy (1860) which is still the subject of discussion by Historians today.

A comprehensive list of the historiography of the renaissance would be pretty lengthy, but i'll list a few titles I have found to be informative.

Books:

Burckhardt, Jacob, The Civilization of the Renaissance in Italy, (London, 1990). - A reprint of Burckhardt's seminal work.

Baron, Hans, The Crisis of the Early Italian Renaissance, (Princeton, 1966).

Pocock, J. G. A., The Machiavellian Moment: Florentine Political Thought and the Atlantic Republican Tradition (Princeton, 2003).

Articles:

Baron, Hans, ’ Burckhardt’s “Civilization of the Renaissance” A Century after its Publication’, in Renaissance News, Vol.13, No.3, (Autumn, 1960).

Hankins, James, ‘The “Baron Thesis” after Forty Years and Some Recent Studies of Leonardo Bruni’, in The Journal of the History of Ideas, Vol.56, No.2, (April, 1995).