European regnal naming conventions: why did European monarchs choose from such a small pool ofnames? England had 8 kings named Henry and 8 named Edward; France had 28 (!) Louis’ and 10 Charles’ - was it simply a popular custom to select a name of a previous monarch to add legitimacy to one’s reign?

On the English side at least, there weren't all that many first names in common circulation in the first place, especially among the aristocracy. There was also a feedback effect where the names of recent or particularly well-regarded monarchs and their consorts got consequently boosted in popularity. And there was a more direct feedback mechanism from the cultural practice of naming sons (especially first sons) for their fathers or particularly distinguished distant anscestors.

There's a joke in the second series of the British sitcom "Blackadder" where Queen Elizabeth I (portrayed in the series as a ditzy, childish sociopath) responds to hearing an unusal name by saying "That's a strange name for a girl. Girls are usually called Elizabeth. Or Mary." There's a fair amount of truth behind that joke: during the Tudor period in particular, most notable women seem to named Elizabeth, Mary, Jane, Catherine, Anne, or Frances. And similarly, most notable men in that period seem to be named Henry, Edward, Richard, Thomas, Robert, Francis, Wiliam, or John.

Henry and Edward, in addition to being very common English names, were also boosted by association with famous anscestors. Henry II ruled over the early peak of the family's territorial holdings in France (Normandy, Anjou, Aquitaine/Gascony, Brittany, and the Vexin) and layed the foundations of the English judicial system, Henry III restored order following the Barons' Wars, and Henry V won England's most sweeping victories in the Hundred Years' War and (very briefly) secured tge French throne for his son. Edward the Confessor (no number, since he predated the Norman Conquest) was the patron saint of English royalty, Edward I Longshanks (now best known as the villain from Braveheart) conquered Wales and secured English hegemony over Scotland and Ireland, and Edward III had a particularly long and prosperous reign in addition to winning major victories in France.

Some of the more common English names only got used once for Kings, probably because the kings of those names were remembered as poor figures for emulation. Particularly John Lackland (the "Prince John" of the Robin Hood stories, also known as "John Softsword"), who lost the family's extensive holdings in France and was so unpopular with his barons that he was very nearly overthrown in favor of a French prince, and King Stephen, who divided the kingdom with a civil war that lasted almost his entire reign.

There were also some extremely common English first names that never got used by kings, most notably Thomas (particularly popular in the Tudor period: c.f. Thomas Woolsey, Thomas Cromwell, Thomas Moore, Thomas Howard, Thomas Bolyen, Thomas Seymore, and Thomas Cramner) and Robert. I'm not sure why, so I'll leave that part for others to answer.