Arthur Tudor was Prince of Wales and heir apparent but died 7 years before his father. If he had lived long enough to be crowned, would he have chosen a boring name, or kept his mythically significant “Arthur”?
What were his parents thinking in naming him something so evocative?
Yes, if Arthur had lived England (and Wales) would have had a King Arthur, possibly represented as King Arthur II, and as you have guessed the choice of the name was certainly loaded with mythical (and supposedly historic) significance.
Henry VII, whose genealogical claim to the throne of England was relatively weak, invoked myths and prophesies of a British resurgence rooted in figures like King Arthur to substantiate his claim to the throne and legitimise his rule, at least early in his reign. However, there was some ambiguity between the ‘traditional’, chivalric idea of Arthur which had been popular in Medieval England, and older British legends, rooted in antiquity, which were much more influential in Wales. Henry invoked both, but both were problematic and arguably ineffective, in different ways.
Henry had long been aware of the symbolic importance of prophecy and legend. During the period of where the future king was in exile on the continent, Welsh poets had made much of Harri Tudur’s descent from the mythical Welsh king Cadwaladr, a birthright associated with an Merlinic prophecy his descendants would free the Welsh from their Saxon oppressors. Through Cadwaladr, Henry could also claim descent from Brutus of Troy, who settled in Britain, drove out the giants who had lived there, and divided the land into three kingdoms for his three sons (his eldest got England, a myth used to justify the suzerainty of English kings over their Scottish neighbours). These legends, then, were rooted in understandings of Britain’s classical past, rather than the chivalric, Medieval Arthurian legend we’re more familiar with today.
Henry and his allies used such beliefs to build support for Henry’s bid for the throne, and at Bosworth he fought under the banner of a red dragon, associated with the Merlinic prophesy of the Welsh freeing themselves from Saxon dominion. They were also used to legitimise Henry after his victory: he continued to use the symbol of the red dragon, with banners displaying the device carried at his coronation and the symbol stamped on coins, and established a commission to work with Welsh bards to draw up his descent from the “ancient kings of Brytaine and the Princes of Wales”.
None of this is directly related to King Arthur—funnily enough, the legend of Arthur wasn’t particularly prominent amongst the Welsh in this time. But it was widely believed in England that the Welsh had prophesised the king’s return to rule over Britain, and this is what Henry was drawing upon when he decided to christen his son Arthur. King Arthur had been an extremely popular figure in England, and had been involved in chivalric displays and pageantry by former English kings, reaching a peak in the fourteenth century.
Henry seems to have deliberately invoked this medieval understanding of Arthur when choosing to name his son. He arranged for him to be born and christened at Winchester Castle, a site associated with the Arthur myth; Winchester was also the home of a round table which was believed to be the original table of Arthurian legend, and Henry probably had it repainted especially for the occasion.
This type of depiction, however, was in decline by Henry’s time, with a number of English historians calling the king’s historicity into question; it was bolstered by the publication of Le Morte d’Arthur in 1485, a compilation of pre-existing legends about Arthur, but on the whole the cultural capital of the Arthurian legend was firmly on the wane by the time Henry’s first son was born.
Three court poets wrote poems celebrating the birth of the prince, but they all invoked classical examples of the prince’s ancestry, and downplayed the significance of the name ‘Arthur’. Two of these poets were Italian, and two French; the lack of English celebration of Arthur’s birth may be symptomatic of a deeper sense of ambiguity for Henry’s English subjects over the British legends which Henry invoked. Many of them were, of course, extremely anti-English, associated with the Britons reclaiming the land from their Saxon enemies.
When, in 1486, the rebellious town of Worcester planned a pageant to welcome Henry into the town (and plead for mercy), they welcomed him by invoking the comparison to 'Arture the very Britan kyng', and Cadwaldr, that famous enemy of the Saxons. Acording to Philip Schwyzer, this represents an “anxious negotiation” of the Welsh legends Henry deliberately invoked, a concern over his association with vengeful anti-English sentiments, and an invitation to the new king to assert his right to rule through his British ancestry but to denounce the more violent associations of this myth.[1]
Not all understandings of Arthur were necessarily negative: Prince Arthur himself was welcomed into Coventry in 1498 by a figure representing King Arthur in the city’s choreographed pageantry, and King Arthur was involved in the similar pageantry which welcomed the prince’s betrothed, Katherine of Aragon, into London in 1501. However, if Henry attempted to legitimise his rule through invoking Arthurian legend, it was not thoroughly embraced by his court or by the English aristocratic elite; David Carlson argued that the myth of Arthur was useful propaganda for the urban middle classes, but was less popular amongst the aristocracy or the educated elite.[2]
Interestingly, the legend of Arthur was occasionally invoked over the rest of the ‘Tudor period’ for political purposes. The supposed example of Arthur, 'Britanniae, Galliae, Germaniae, Daciae imperator', was used to justify the imperial authority of Henry VIII, both at home and abroad, a move which seems to have provoked scorn, especially from representatives of the Holy Roman Emperor of the time, Charles V. At some point before Charles’s visit to England in 1522, the supposed round table at Winchester was repainted, with the Tudor rose pointedly placed at its centre. And later in the century, the occultist John Dee used the example of the emperor Arthur to justify Queen Elizabeth's claims to lands in the Atlantic, an attempt to prompt the queen into more vigorous colonial projects.
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[1] Philip Schwyzer, Literature, Nationalism, and Memory in Early Modern England and Wales (Cambridge, 2004), pp. 22 – 25.
[2] David Carlson, ‘King Arthur and court poems for the birth of Arthur Tudor in 1486’, Humanistica Lovaniensia, Vol. 36 (1987), pp. 165 – 166.
Just to add beside /u/SomewhatMarigold's excellent answer: historically, it was not the norm for English/British monarchs to change their name on assuming the throne. As I discussed in this past answer, Queen Victoria appears to be the first to have done it (her real name was Alexandrina Victoria), followed by her son, whose name was Albert Edward; he skipped his first name as well, claiming to be honoring his father. George V and Edward VIII came next, using their actual first names, and then came George VI, whose name was Albert Frederick Arthur George: I've never seen a definite explanation given, but most likely he wanted to emphasize continuity with his father, George V, because Edward the Abdicator was between them.