Why "Giraffe" as a name for the animal?
From the OED, Camelopard was first recorded in
▸ a1398 J. Trevisa tr. Bartholomaeus Anglicus De Proprietatibus Rerum (BL Add. 27944) (1975) II. xviii. xx. 1159 Cameleopardus hatte cameleopardalis also... And haþ þe heed of a camele..and spekkes of þe parde. [And has the head of a camel and the spots of the leopard.]
Somewhere between then and 1594 we have early records of "Giraffe":
1594 T. Blundeville Exercises v. ix. f. 259 This beast is called of the Arabians, Gyraffa. 1617 F. Moryson Itinerary i. iii. v. 263 Another beast newly brought out of Affricke..is called..Giraffa by the Italians.
β. c1600 Sanderson in Purchas Pilgrims (1625) ii. 1619 The admirablest and fairest beast that euer I saw, was a Iarraff.
[Although there was obviously some confusion as the animal was so rare in the UK:
1688 R. Holme Acad. Armory ii. 130/2 Beasts..Such as chew the Cud, and are not Horned, as Camelopard Giraffa.]
It seems from the Google Ngram links above, that the name started to change decisively from Camelopard around the end of the 18th century and the move to Giraffe became all but complete around 1824. This was probably a cultural change that followed the scientific advances in which "Camelopard" was seen as a crude and primitive combination of exterior features, whereas "modern people" required something a little more specific, and hence the Arabic.
Thus we have in The Kaleidoscope: or, Literary and scientific mirror, Volume 8 1821 p81
The ROMANS, to whom Julius Caesar was the first who introduced a giraffe, called the animal camelo-pardalis or camelopard. But a multitude of essential differences distinguish it from the camel; and of the leopard, […] it must also be observed that they [the spots] differ in being flat and irregular, instead of being round and ranged en rose. The ancient name of Zerapha, corrupted by us into giraffe, is much more fitting, therefore, that that which the Romans substituted for it.
As far as other names are concerned, there was apparently one:
1605 J. Sylvester tr. G. de S. Du Bartas Deuine Weekes & Wks. i. vi. 194 Th' horned Hirable [1605 marg. alias, Girafle, 1608 marg. Alias, Gyrafa]
But this may be a mistake of the translator, hence the marginal notes.
In Middle English, camelioun (via Medieval Latin) and (sometimes) gerfaunt and orafle (both from Old French) referred to the giraffe (Middle English Dictionary). Authors with more knowledge of Latin would also know cameleopardalus directly, though this word was usually glossed in context until the sixteenth century (OED). For English audiences, the animal now denoted by giraffe was mainly a creature from bestiaries or travelogues, and not one they would have direct experience with. That changes over the next few centuries.
I'll split this in two. The first part casts giraffe's early history (including, maybe, why we spell it giraffe - thanks, Italy!). The second part delves into when the word became more popular than camelopard.
Giraffe from Italian to English
Giraffe comes into the picture in the late 16th century, in a period when many multilingual lexicons are being produced and new words from Latin, Spanish, French, and Italian are flooding into the language. The giraffe, as a word, is imparted into English through a succession of translations from Romance languages that describe encounters with the wider world. Abraham Hartwell uses it in his translation of The History of the warres between the Turkes and the Persians (1595):
a lyue Giraffle: (which is a beast like a Cammell and a Panther,)
Note the use of the newer term (giraffle, possibly similar to the Italian original) with the description that combines two animals (camel + panther ~ camel + leopard). John Florio defines the Italian version of giraffe only a few years later in his World of Words (1598):
Giraffa → , a beast greater then a leopard, resembling a panther with a long neck. Also a kinde of glasse that they vse to drinke wine in, in Italie.
Again, the panther descriptor sticks: "a panther with a long neck." Side note: Florio gives no cross-reference whatsoever to another word similar in meaning, possibly indicating that Florio thought giraffe was different from camelopardo and thus worth defining on its own:
Camelopardo,a beast begotten of a ca∣mell and a panther.
In 1600 giraffe appears in John Pory's translation of A Geographical History of Africa by Arabic author Leo Africanus:
Of the beast called Giraffa.
THis beast is so sauage and wilde, that it is a very rare matter to see any of them: for they hide themselues among the deserts and woodes, where no other beasts vse to come; and so soone as one of them espieth a man, it flieth foorthwith, though not very swiftly. It is headed like a camell, eared like an oxe, and footed like a *: neither are any taken by hun∣ters, but while they are very yoong.
The British Library describes this book as an important influence on early modern perceptions of Africa. The translator preserves the giraff- spelling. Another famous book on the history of the Turks came out a few years later by Richard Knolles (1603). Writing on his own, rather than translating another document, he uses giraffe:
a liue Giraff•e (which is a beast like a Cammell and a Panther)
He's copying Abraham Hartwell in this section of the book, with slight updates to the spelling. This history was very popular and would be read widely in the next couple of centuries (Britannica cites Samuel Johnson and Lord Byron as fans of Knolles).
So it's likely, based on these early uses, that the spelling giraffe (especially the double-f) comes from early translations from Italian and subsequent copying of those Italian spellings. We can thank Hartwell, Florio, Pory, Knolles, and perhaps others for that spelling.
Whence giraffe over camelopard
Throughout the 17th and 18th centuries, giraffe and camelopard both appear a lot. In EEBO-TCP (up to 1700), giraf- appears 80 times and camelo- 87 times. (The wildcard searches help find alternate spellings.) In ECCO (1700-1800), giraffe appears 109 times and camelopard appears 135 times.
In the 19th century, something changes. An N-gram from commenter user121863 shows some kind of shift early in the century. This shift is visible in academic texts: JSTOR turns up 28 results for camelopard in the 19th century, and 459 results for giraffe. In particular, many of the results for giraffe represent a naturalist and anatomical bent toward studying the animal, with firsthand accounts and more accurate visual depictions taking precedence. So in the same issue of the Dublin Penny Journal (Feb. 6, 1836), we have an article on ruminating animals that only mentions the giraffe:
The giraffe, again, is confined entirely to Africa.
Then there's an entire article that mentions both the camelopard and giraffe and uses both terms in the article body:
THE CAMELEOPARD, OR GIRAFFE. ... The height of the giraffe is from fifteen to twenty-one feet. ... The cameleopard was seen by Denham and Clapperton in parties of five or six on the borders of Lake Tehad ..."
It's hard to pinpoint a moment when cameleopard gave way to giraffe, but this concession of using them together was less and less common in the period. Many texts by 1836 already use giraffe alone. The shift to giraffe is well underway.
I ran a series of searches of the Early English Books Online database to see how frequently various names and spellings of the creature in question were used in unique instances (that is, excluding subsequent editions of certain popular works) in English-language texts (that is, excluding foreign-language texts and dual-language dictionaries that simply define words from one language in the other language) during the period 1475–1700. Here are the results, ordered by date of earliest occurrence, with spelling, number of unique occurrences, and years of occurrence identified for each:
cameloparde: 2 occurrences (1567 and 1609–1610)
cameleoparde: 1 occurrence (1572)
cameleopard: 6 occurrences (1577, 1601, 1671 1695, 1697, and 1699)
camelopardalis: 16 occurrences (1580{?}, 1594, 1598, 1607, 1612, 1613, 1627, 1634, 1652, 1653, 1656, 1670, 1678, 1682, 1686, and 1694)
cameleopardalis: 2 occurrences (1592 and 1634)
gyraffa: 6 occurrences (1594, 1607, 1611, 1652, 1670, and 1675)
giraffle: 1 occurrence (1595)
giraffa: 14 occurrences (1598, 1600, 1607, 1613, 1617, 1630, 1634, 1657, 1668, 1678, 1680, 1682, 1686, and 1688
gyrapha: 1 occurrence (1607)
hirable: 2 occurrences (1611 and 1679)
garaffa: 2 occurrences (1625 and 1693)
iarraff: 1 occurrence (1625)
giraffe: 7 occurrences (1634, 1635, 1660, 1677, 1679, 1687, and 1692)
camelopardalus: 3 occurrences (1634, 1657, and 1693)
camelo-pard: 3 occurrences (1645, 1658, and 1686)
jaraff: 1 occurrence (1657)
girafe: 3 occurrences (1660, 1679, and 1677)
camelopard: 3 occurrences (1668, 1670, and 1688)
camelopardelis: 1 occurrence (1670)
camelo pardalis: 2 occurrences (1675 and 1678)
cameleo pardalis: 1 occurrence (1678)
geraffe: 1 occurrence 1679
gyraffe: 1 occurrence (1679)
camel[-]leopard: 2 occurrences (1686) and 1688
came-leopard: 1692
Altogether, in English-language books in the EEBO database, I count 43 unique occurrences of references to members of the cameleopard family, 38 unique occurrences of members of the giraffe family, and 2 references to the hirable. The earliest mention of a name from the first family of spellings (cameloparde) is from 1567, and the earliest mention of a name from the second family of spellings (gyraffa) is from 1594. Both the cameleopard family of names and the giraffe family of names were still being used in published texts as the seventeenth century drew to a close.
These two points suggest first that the terms came into English (as opposed to the Latin of learned Englishmen) not so very many years apart, and second that the victory in English of giraffe over cameleopard was not swift.