Although I realise there is probably a wealth of academic literature on him, Henry seems to be chiefly known for reigning forever and having a zoo, whilst his nemesis Simon De Montfort seems to hold more of a grip on imagination. So why is this King lacking in popularity in modern eyes? Was he incompetent but not as incompetent to be as memorable as his father King John, or was he greatly overshadowed by his son Edward I?
A little bit of all of these things. Henry III wasn't really a remarkable figure. His regent was one of the most famous knights of the middle ages, William Marshal. His dad was one of the most maligned kings in English history. His son was a great administrative reformer, suppressed Wales, went of crusade, beat Simon de Montfort, and was involved in well-known battles with Scotland. What did Henry do? He reissued Magna Carta a couple of times. His mother, Isabelle of Angouleme, goaded him into (disastrously) trying to reclaim Normandy and then sent his spoiled half-siblings from Lusignan to England where Henry funded their lavish lifestyles - this made him very unpopular. Henry was not a good diplomat, he was not a political reformer, and he did not want to get into another fight with the Pope. Henry was pious and liked Edward the Confessor. Even though he reigned for over fifty years, if there's nothing really exciting going on, it is hard for a historian to justify writing a monograph length work on him. Even academic works are sparse on his reign, though he is always included when works come out on the Angevins, Plantagenets, or England in the High Middle Ages.
Here are a couple of works that provide further reading:
(1) David Carpenter. The Minority of Henry III (1990)
(2) Bjorn Weiller. Henry III of England and the Staufen Empire, 1216-1272 (2006)
(3) Robert Stacey. Politics, policy, and finance under Henry III, 1216-1245 (1987)
(4) Nicholas Vincent. The Holy Blood : King Henry III and the Westminster Blood relic (2001)
Hope this helps a little. Happy Reading!