Others will pull out the traditional argument and cite Edward V‡ or Henry VI part II¶. Both Matilda§ and Stephen◊ have their supporters and there are those who will look back even further, citing Canute¥ or Partacanute℘.
Even Arthur∅ and Richard IIℑ have their supporters, despite both being immortalised in the media. Oh, and who can forget the odd Jacobite who will insist on Charles IIIℵ.
But, I think, the most popular answer will be Lady Jane Grey∇ (although some would argue that the most famous obscure monarch⊗ is in fact Æthelred the Unready♠).
Despite all of this, SGO research historians have recently discovered two even more obscure monarchs — Edgar the Atheling♥ and Louis♦.
∗ Famous for having a red beard, and dying on the hunt.
† Who, despite the impression Robin Hood stories tend to give, actually was King for some years after Lionheart's death.
‡ Famous for being locked in the Tower and killed, although noöne can agree by whom.
§ Or is it Maud?
◊ "this and more we suffered nineteen winters for our sins."
¥ Subject of an early attempt at spin involving oceans.
℘ Fabricated out of whole cloth by Sellar and Yeatman.
∅ Famous for mixing burnt cakes and wine in the cup from our Lord's last supper.
ℑ Invented by Shakespeare.
ℵ Who unfortunately fails on both counts, being neither obscure nor a monarch.
∇ Around whom we appear to have built a kind of underground national history cult.
⊗ A pleasingly oxymoronic phrase.
♠ Who was eventually expelled from the Danish Jazz group for taking too long to set up his clarinet.
♥ Who succeeded Harold after the Battle of Hastings, and later abdicated to William the Conqueror before he was crowned.
♦ Who kind of foreshadowed Williamandmary♣.
♣ One of the main reasons for Footnote 11.