When was the New Kingdom of Egypt (c. 1550BC to 1077BC) first called "New Kingdom" (or something cognate)?
According to the Oxford Companion to Archaeology, the term New Kingdom was introduced by the German historian Eduard Meyer in his Geschichte des Altertums. English Egyptologists adopted his usage, displacing their earlier designation of the period as the "Empire".
Source: Silberman, Neil Asher, and Alexander A. Bauer, eds. The Oxford Companion to Archaeology. Oxford University Press, 2012.
The division of Egyptian history into Old and New Kingdoms was pioneered by Carl Richard Lepsius. In his 1858 Königsbuch der alten Ägypter, Lepsius grouped Manetho's Dynasties under the categories Alten Reich and Neues Reich, assigning Dynasties XVII to XXXI in the latter. This periodisation seem to have been inspired by Baron von Bunsen's work in the 1830s or 40s.
I'm not sure why the Oxford Companion to Archaeology chose to credit Meyer over Lepsius here.
According to this:
the designation “Neues Reich” was in use since 1834. This was about 20 years before the birth of Eduard Meyer.
The first publication I know of was by Christian Charles Josias, the Baron von Bunsen, in his Aegyptens Stelle in der Weltgeschichte (1845). This was the publication that brought the classification into general use among English-speaking Egyptologists. To quote from Browne's Hierogrammata:
The tract of time from Menes downward is divided by M. Bunsen and Lepsius, into three portions, which they designate by the names, Old, Middle, and New Kingdom. The first of these ends at or shortly after the close of the great Diospolitan Dynasty of the Osortasens, or Sesortesens and Amenemhes, which in Manetho is the Twelfth Dynasty. The third, or New Kingdom, begins with the Diospolitan Dynasty of Amosis, which is Manetho's Eighteenth. The intermediate period, that of the Middle Kingdom, is the time of the usurped dominion of the Hyksos or Shepherd-Kings.
The epoch of the New Kingdom is placed by M. Bunsen at 1688 B.C. The grounds on which this determination rests will be fully examined in the Second Chapter of this Work, where it will be shewn, by monumental and documentary evidence, that the Eighteenth Dynasty began about fourteen Centuries, only, before Christ.