Henry III did in fact name his heir (Edward I) after his favorite saint, Edward the Confessor.
...the King was particularly devoted to the figure of Edward the
Confessor, whom he adopted as his patron saint.
Of course there may have been some secular calculation behind this. It appears to have been at least partly a decision to take his rulership of England seriously after a difficult rebellion earlier in his reign.
The events of the civil war in Henry's youth deeply affected him, and
he adopted Anglo-Saxon king Edward the Confessor as his patron saint,
hoping to emulate the way in which Edward had brought peace to England
and reunited his people in order and harmony. Henry tried to use
his royal authority leniently, hoping to appease the more hostile
barons and maintain peace in England.
The Civil War being referred to here was the First Baron's War, which among other things gave us the Magna Carta.
cute video here
Basically, typical medieval land shenanigans had left the Kings of England also owning more land in France than the King of France. This repeatedly tempted them to take over France (generally failing). The English barons had enough of this behavior (and the taxes on them paying for it), and rose up in revolt in 1215. That allowed the French Heir (the future Louis VIII) to put the shoe on the other foot, and invade England in support of the Rebels. The military campaigning generally went very badly for the English king, with the cherry on top being him dying of dysentery.
This left this whole mess to his heir Henry III, at the ripe old age of 9.
Young Henry's advisors managed to turn the war situation around. Mostly the rebel Barons were just mad at his father, and as nobles themselves they had sympathy and respect for an argument against stealing an underaged noble's inheritance. They also likely saw more virtue in the monarchy being run by Henry's loyal advisors, who were largely English barons like themselves, rather than being run by the future King of France.
However, his attempts to get back his lost lands in France didn't fare so well, and of course while he was gone a fresh rebellion broke out, which again he was forced to settle mostly outside the field of battle. At this point his situation was this: unlike his father who had vast French domains and visions of a European empire, Henry found himself King of a very suspicious England, and a much more minor baron himself in France.
So you can see where a ruler in this particular situation might see great value, particularly politically, in playing up his Englishness.