They did so because it was the actual general use of the term.
"Palestine" was and is more a geographical designation, since Roman times, while "Israel" was and is a politically one and "Holy Land" was and is a spiritual-religious one.
The Balfour declaration was a bit vague and aspirational and also focusses on defining the geographical location.
PALESTINE, a geographical name of rather loose application. Etymological strictness would require it to denote exclusively the narrow strip of coast-land once occupied by the Philistines, from whose name it is derived. It is, however, conventionally used as a name for the territory which, in the Old Testament, is claimed as the inheritance of the pre-exilic Hebrews; thus it may be said generally to denote the southern third of the province of Syria. Except in the west, where the country is bordered by the Mediterranean Sea, the limit of this territory cannot be laid down on the map as a definite line. The modern subdivisions under the jurisdiction of the Ottoman Empire are in no sense conterminous with those of antiquity, and hence do not afford a boundary by which Palestine can be separated exactly from the rest of Syria in the north, or from the Sinaitic and Arabian deserts in the south and east; nor are the records of ancient boundaries sufficiently full and definite to make possible the complete demarcation of the country. Even the convention above referred to is inexact: it includes the Philistine territory, claimed but never settled by the Hebrews, and excludes the outlying parts of the large area claimed in Num. xxxiv. as the Hebrew possession (from the “River of Egypt” to Hamath). However, the Hebrews themselves have preserved, in the proverbial expression “from Dan to Beersheba” (Judg. xx. 1, &c.), an indication of the normal north-and-south limits of their land; and in defining the area of the country under discussion it is this indication which is generally followed.
Taking as a guide the natural features most nearly corresponding to these outlying points, we may describe Palestine as the strip of land extending along the eastern shore of the Mediterranean Sea from the mouth of the Litany or Kasimiya River (33° 20′ N.) southward to the mouth of the Wadi Ghuzza; the latter joins the sea in 31° 28′ N., a short distance south of Gaza, and runs thence in a south-easterly direction so as to include on its northern side the site of Beersheba. Eastward there is no such definite border. The River Jordan, it is true, marks a line of delimitation between Western and Eastern Palestine; but it is practically impossible to say where the latter ends and the Arabian desert begins. Perhaps the line of the pilgrim road from Damascus to Mecca is the most convenient possible boundary. The total length of the region is about 140 m.; its breadth west of the Jordan ranges from about 23 m. in the north to about 80 m. in the south. According to the English engineers who surveyed the country on behalf of the Palestine Exploration Fund, the area of this part of the country is about 6040 sq. m. East of the Jordan, owing to the want of a proper survey, no figures so definite as these are available. The limits adopted are from the south border of Hermon to the mouth of the Mojib (Arnon), a distance of about 140 m.; the whole area has been calculated to be about 3800 sq. m. The territory of Palestine, Eastern and Western, is thus equal to rather more than one-sixth the size of England.
There is no ancient geographical term that covers all this area. Till the period of the Roman occupation it was subdivided into independent provinces or kingdoms, different at different times (such as Philistia, Canaan, Judah, Israel, Bashan, &c.), but never united under one collective designation. The extension of the name of Palestine beyond the limits of Philistia proper is not older than the Byzantine Period.
–– Britannica 1911.
The British were not alone in this, as the Zionists lobbying for such a creation at the time did so likewise:
Zionism (Hebrew: צִיּוֹנוּת Tsiyyonut [t͡sijo̞ˈnut] after Zion) is the nationalist movement of the Jewish people that supports the re-establishment of a Jewish homeland in the territory defined as the historic Land of Israel (roughly corresponding to Canaan, the Holy Land, or the region of Palestine). Modern Zionism emerged in the late 19th century in Central and Eastern Europe as a national revival movement, both in reaction to newer waves of antisemitism and as an response to Haskalah, or Jewish Enlightenment. Soon after this, most leaders of the movement associated the main goal with creating the desired state in Palestine, then an area controlled by the Ottoman Empire.
As the Jewish Virtual Library asserts:
A derivitave of the name Palestine first appears in Greek literature in the 5th Century BCE when the historian Herodotus called the area Palaistin?(Greek - Παλαιστ?νη). In the 2nd century CE, the Romans crushed the revolt of Shimon Bar Kokhba (132 CE), during which Jerusalem and Judea were regained and the area of Judea was renamed Palaestina in an attempt to minimize Jewish identification with the land of Israel.
Under the Ottoman Empire (1517-1917), the term Palestine was used as a general term to describe the land south of Syria; it was not an official designation. In fact, many Ottomans and Arabs who lived in Palestine during this time period referred to the area as Southern Syria and not as Palestine.
After World War I, the name Palestine was applied to the territory that was placed under British Mandate; this area included not only present-day Israel but also present-day Jordan.
Leading up to Israel's independence in 1948, it was common for the international press to label Jews, not Arabs, living in the mandate as Palestinians. It was not until years after Israeli independence that the Arabs living in the West Bank and Gaza Strip were called Palestinians.
The word Palestine or Filastin does not appear in the Koran. The term peleshet appears in the Jewish Tanakh no fewer than 250 times.
And arguably this regional designation slowly took on an ever better defined state through the 19th century. But it can be said assuredly that the word 'Palestine' for the general region is not a British invention, neither from the encyclopaedia of 1911 already quoted, nor in an older edition from 1859, nor in the Balfour declaration of 1917.
How the modern boundaries began to take shape for that region?
The process of determining Palestine’s boundaries in the modern era started in the first half of the nineteenth century, and has not yet ended. Different political actors operated during different periods in designing the borderlines and these actions led to the unending constant process of changing the land’s boundaries. This process gave birth to the term ‘land of many boundaries’, which is the subject of this book. It started out with the setting of an internal line within the Ottoman Empire, in the early 1840s.
–– Gideon Biger: "The Boundaries of Modern Palestine, 1840–1947", Routledge: London, New York, 2004.
As such the term Palestine referred to a bigger geographical region than 'the Holy Land', in most eyes. 'The Holy Land' would also have been seen as also taking sides on religous grounds, but would at the same time for Jews and Muslims center around Jerusalem and little else, for Christians including Galilee of course. That would then conform to ancient history divisions like for example the smallish state of Judah or the province Yehud, but certainly not any mythical imagination like 'King David's empire'. And since boundaries are always arbitrations the declaration opted to declare to support a homeland for Jews in Palestine. That leaves open easily that this homestead might be a tiny part of the region, or comprise the entire region or whether it actually contains Jerusalem or not.
As it was the political and spiritual goal of crusaders to get to 'the Holy Land', I won't comment on that. But it seems I need to on Mark Twain.
What did Mr Clemens write about 'the Holy Land' in his book about the subject matter?
If ever those children of Israel in Palestine forget when Gideon's Band went through there from America, they ought to be cursed once more and finished. It was the rarest spectacle that ever astounded mortal eyes, perhaps.
–– Mark Twain: "The innocents abroad, or, The new Pilgrim's progress", being some account of the steamship Quaker City's pleasure excursion to Europe and the Holy land ; with descriptions of countries, nations, incidents and adventures, as they appeared to the author
Twain, Mark (1835-1910)
This level of ambiguity in the Balfour declaration is probably seen in diplomatic circles as a good thing™.