The Kingdom of France was a continuation of the Frankish Kingdoms just renamed, so would a French king named 'Clovis' become 'Clovis IV' if he ascended to the the throne post-Merovingian times? I know that Charlemagne is considered a king of France, and he inherited the title from his father who was most certainly a king of the Franks, which makes me think that the Merovingian would be considered French kings.
It really wasn't a name change so much as a change in historiography. At a certain point, usually the accession of the Capetians in 987, modern historians view the Kingdom of the West Franks as being France. The title Francorum Rex was used by French kings long after we would consider them to be king of France, well into the Middle Ages. Also, the Franks began using Latin and Gallic Romance. So there really is not clear line between Francia and France in terms the polity that was actually there, so much as historians making a line between the West Frankish Kingdom that emerged from the Carolingian empire and the medieval French kingdom. Phillip II Augustus, who ruled from 1180-1223, was the first to adopt the title "King of France" rather than "King of the Franks", about 10 years into his reign.
Sources: