Why did Henry III of England give his sons English names, other than naming them after his favourite saints?

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.

Although T.E.D fundamentally answered your question, I would like to share some insight on Henry III's devotion to Edward the Confessor for which one can say that was the core motive for naming his son the unusual name Edward in the 13th century. As @T.E.D already said the First Baron's War impacted the young Henry immensely, but here are the motives behind his actions for naming his son Edward from a different angle:

But Henry III had good reason for foisting this unfashionable name on his firstborn son. After his father’s death, his mother had abandoned him – Isabella of Angouleme left England for her homeland in France, remarried and never returned. Effectively orphaned from the age of nine, the young king had found substitute father figures among the elderly men who had helped him govern his kingdom. But these men too, Henry ultimately decided, had failed him, and by 1234 he found himself alone once more. It was at this point, though, that the king discovered a new mentor, a man who would never, ever let him down – largely because he had already been dead for the best part of two centuries.

Henry’s new patron was Edward the Confessor, the penultimate king of Anglo-Saxon England. Like Henry himself, Edward had not been a very successful ruler: his death in January 1066 had sparked the succession crisis that led to the Norman Conquest nine months later. Posthumously, however, Edward had acquired a reputation as a man of great goodness – so much so that, a century after his death, he had been officially recognised as a saint. Thereafter his reign had acquired the retrospective glow of a golden age: men spoke with great reverence about his good and just laws (even though, in reality, he never made any). Of course, the fact that Edward was not a great warrior had made him an unlikely exemplar for the conquering dynasty of kings who came after him. But to a man like Henry III, who was entirely lacking in military skill, the Confessor seemed the perfect role model. There were, moreover, other similarities between their two lives that must have struck Henry as highly significant. Edward had lost his father and been abandoned by his mother at a young age; he had grown up with war and wished to cultivate peace; he had been misled by treacherous ministers. Above all, Edward, like Henry, was famed for his piety.

Edward was the king who established the royal palace at Westminster, in order to be near the great abbey (minster) that he spent the last years of his life rebuilding. In due course he was buried in the abbey church, and his tomb there became a pilgrim shrine. It was the greatest testament to Henry III’s love and reverence for the Confessor that, from 1245, he would spend vast sums rebuilding the abbey for a second time, replacing the old Romanesque church with the massive Gothic building that stands today. It was no surprise to anyone, therefore, that Henry should choose to call his son Edward in honour of his idol.

As for the problem on his name it is said:

Towards the end of Edward I’s reign, for example, some of his subjects felt compelled to chronicle his remarkable deeds, and decided that they needed to distinguish the king by giving him a number. Unfortunately, they miscounted, including in their tallies the Confessor (who ruled from 1042 to 1066), and also the celebrated tenth-century king, Edward the Elder (899–924), but overlooking entirely the short and unmemorable reign of Edward the Martyr (975–78). For this reason, at least two thirteenth-century writers referred to Edward I as ‘Edward the Third’. Had they counted correctly, they would have called him ‘Edward the Fourth’. Fortunately for us, such early and inaccurate numbering schemes did not endure. In general, when his contemporaries wished to distinguish Edward, they called him ‘King Edward, son of King Henry’. The need for numbers arose only after his death, when he was succeeded by a son, and then a grandson, both of whom bore his illustrious name. By the middle of the fourteenth century, Englishmen found themselves having to differentiate between three consecutive, identically named kings, and so unsurprisingly they started referring to them as the First, Second and Third. Anyone troubled by the recollection that once upon a time there had been other kings called Edward could salve their historical conscience by adding ‘since the Conquest’. Thus the Norman Conquest became the official starting point for the numbering of English kings. But it was only necessary to have such a starting point in the first place because of Henry III’s idiosyncratic decision to resurrect the name of a long-dead Anglo-Saxon royal saint and bestow it on his eldest son.

Source: A Great and Terrible King: Edward I and the Forging of Britain by Marc Morris