A useful source book for those interested in Anatolian toponymy is Demetrius J. Georgacas's The Names for the Asia Minor Peninsula published in 1971. Despite some minor errors here and there (see below for an example), it has lots of data and lists of references to be used as starting points for further research.
One interesting question is when the toponym Turkey, or any of its variants, was first used for the general geographical area where the country of Turkey is now located. According to Georgacas, the first recorded uses of the name Turquie in French works date to the end of the 12th and the beginning of the 13th centuries. He states that the first known uses in English come a little later: "English Turkye occurs in Chaucer (ca. 1369) and secondly in the form Turkey in Sir John Maundeville (1371)."
I was able to find 2 late 19th century editions of The Voiage and Travayle [Travaile] of Sir John Maundeville Knight in Google Books. The older one, first published in 1839 and reprinted in 1883, was actually a reprint of a 1725-7 edition based on an early 15th century manuscript. The 1839 reprint, luckily for us, retained the original language and was annotated by J. Q. Halliwell. Here is the page where Maundeville first mentions Turkye, a spelling surprisingly similar to the modern Turkish Türkiye.
Maundeville's statements "the gate of Civetot (=Chienetout)...the hill of Civetot...is a mile and a half from Nyke" explains what part of Turkey he was talking about. According to Foss*, Civetot was the Latin name of Kibotos or Helenopolis (the present day Hersek) on the southern coast of the Izmit Bay near the eastern end of the Sea of Marmara. Nyke must have been Nicaea (present day Iznik) ~22 miles southeast of Civetot and the gate on a hill, referred to by Maundeville must have been the castle known as Xerigordos. Foss describes its location as follows: "It stands above the valley of the [river] Drakon and the road from Helenopolis to Nicaea...about five miles south of the promontory of Helenopolis."
In the later, 1887 edition, the editor John Ashton "translated" the text into contemporary English and Turkye became Turkey.
Note that Ashton's tentative identification of Nyke as Salonika is groundless.
Georgacas cites Ashton's 1887 version of Maundeville as his source for the use of the spelling Turkey. He obviously didn't realize that Ashton had changed the spellings of many of the words in the text and that the original spelling at least in the 15th century version of Maundeville was Turkye, not Turkey.
Turkye appears to have meant "the land of Turks" and was not a distortion of some other older word. This is clear from Maundeville's description (in the original English) of what he saw at Ephesim (Ephesus), which clearly associates Turkes with Turkye.
His last 2 sentences translate as: "And Turks hold now all that place and the city and the church. And all Asia the less [Asia Minor] is y cleped Turkye". Ashton translates the last sentence as "& therefore is Asia the less called Turkey".
Incidentally, I don't know the meaning of Maundeville's statement "that is clept Aungeles Mete" in reference to the "manna" from St. John's tomb. Interestingly, Ashton took out that sentence. Foss, in Pilgrimage in Medieval Asia Minor, discusses St. John's tomb in Ephesus, but there is nothing there that would explain Maundeville's words.
As Ahton discusses in the introduction to his edition of Maundeville, there were apparently some doubts as to whether Maundeville was a real person or if he was, whether he really went on the travels he claimed he had gone. This is, however, a moot point as fas as the use of the 14th or 15th texts ascribed to Maundeville as toponymical sources.
*Clive Foss (1996) Survey of Medieval Castles of Anatolia II Nicomedia.