(This post owes a great debt to this recent conversation between selenak and myself. Please note that many of the ideas in it are hers.)
Theory one: Morgause is actually Arthur’s half-sister too; they are both Ygrain's children. She’s also Morgana’s half-sister through their father Gorlois.
There are actually two ways this could work: Morgause could be the daughter of Ygrain through a previous relationship with Gorlois, who went on to become Morgana’s father, or Ygrain could have had an affair with Gorlois while married to Uther.
• Don’t you know that on television hair color is always significant? The blondes and blond must be related! In other words, they did not have to cast Arthur, Morgause, & Ygrain as they did.
• Morgause tells Arthur she knew his mother well.
• It would explain why Morgause and Morgana are only half-sisters.
• It would explain why Morgause doesn’t mean any threat to Arthur the first time she meets him, in 2x08, but waits to test him before turning on him.
• Arthur and Morgause’s skill with swords could be hereditary, through Ygrain.
• The show has already set up Arthur and Morgause as parallel characters in many ways—including their mysterious births (both of which involve Gaius swearing a solemn oath never to reveal the truth).
• If we work with the Ygrain-had-an-affair theory, it would explain
- why Morgause was born in Camelot (Ygrain lived there),
- why she was smuggled out and given to priestesses to raise, given that she was born a good ten years before Uther turned against magic,
- and the rumors that Uther sent his good friend Gorlois to his death.
• Ygrain being unable to conceive a child except with magic is too important to the world-building to circumvent.
• How well could Morgause have known Ygrain if she was smuggled out of Camelot shortly after her birth? We have many reasons to suspect her honesty.
• Morgause is never as tender or protective toward Arthur as she is toward Morgana. In fact she turns on him awfully quickly after he fails to kill Uther, and clearly intends for the Knights of Idirsholas to kill him (unlike Morgana).
• Ygrain’s brother Tristan DuBois does not seem to have been a great swordsman—Uther beat him twice, and Pellinore and even Owain would have killed him if not for the magic—so the fighting-skills argument is weak.
• The show has already gone to great lengths to remove the adultery element from Arthur’s magical birth, and probably does not want to present its young viewers with a plot that involves it.
• If we avoid adultery, we need a divorce, but which the show is likely to avoid as a false anachronism (despite the fact that medieval divorce was not uncommon).
Theory two: Morgana is actually the daughter of Uther, not Gorlois. She is therefore half-sister to both Arthur (through their father) and Morgause (through their unnamed mother).
• A conversation in 2x03 deliberately hints this. Uther says Morgana “takes after her father too much,” and stiffens when Gaius mentions Gorlois and says Uther has “looked after that girl as if she were your own.”
• It would explain why Uther became Morgana’s guardian but never treated her as a prospective bride for Arthur (unlike Arthur, Gwen, and Morgana herself).
• It would explain the rumors Morgana alludes to in 1x12 that Uther sent her father to his death, and his reaction when she brings them up. (Actually, it would cast all of 1x12 in an interesting light.)
• It would explain Uther’s increasing obsession with Morgana’s safety (which began shortly after the writers squelched the Arthur/Morgana romance).
• It would explain why Morgause and Morgana are only half-sisters.
• It would be completely consistent with Uther’s character to cover such an affair up, add a new dimension to his guilt in 2x12, and give him a reason to hide Morgana’s sister from her.
• Morgana and Uther are very similar characters, with the same tendencies toward denial, personalizing all conflicts, and vengeance.
• If they don’t share a father, why would Morgause give Morgana a bracelet with the crest of Gorlois?
• The same objection to adultery on a kids’ show still stands, and there’s no way to avoid it in this scenario.
• The troll episode implies that Uther is a one-woman man, and that woman is Ygrain.
• It doesn’t explain why Gaius smuggled baby Morgause out of Camelot and away from her parents years before magic was banned.
• If Uther is Morgana’s father, it’s a tightly-guarded secret only he knows. So why would Gaius openly refer to Morgause as Morgana’s half-sister in front of him?
Theory three: Morgause is the daughter of Uther and a woman with powerful magical powers who then went on to marry Gorlois and bear Morgana, and then died (naturally or not) around the time of the Purge.
• This theory would explain why both Morgause and Morgana have powerful magic. (Both Merlin and Edwin inherited their magic from parents.)
• It would explain why Morgause was raised by powerful priestesses of the Old Religion, while Morgana, born around the Purge, grew up a clueless noblewoman.
• It would explain why Morgause has magic and Arthur doesn’t, but allow them to be related, explaining the parallels between them.
• We already know Uther has a tendency to be attracted to magic and then revile it.
• Uther could have begotten Morgause before he married Ygrain, thereby avoiding adultery and anything age-inappropriate.
• No hair color match, unless Uther really has a thing for blondes.
• Again, it requires Uther to be more of a player than we’ve seen him.
• Again, the bracelet thing.
• Do you really think the coolly cunning Morgause is the daughter of hotheaded Uther? What do they have in common besides swordfighting skills and the desire to rule Camelot?
• (If we follow the second option) Hiding the truth about Ygrain’s death is one thing, but do you really think Uther could hide it if he had killed Morgana’s mother?
Theory four: Morgana and Morgause are half-sisters, but neither is related to Arthur. The mysteries of Morgause’s birth and Uther’s break from Gorlois have other answers, probably involving more magic than sex.
• The solemn oath Gaius swore to never reveal the truth of Morgause’s birth is implied to have something to do with magic and the Old Religion.
• It would fit with the emphasis on magic in the evolution of Morgana & Morgause’s relationship.
• It’s an opportunity to bring Nimue back and explain her role better.
• Probably the most age-appropriate answer, given the show’s target audience.
• Why make things simple? The other theories are much more fun.
If I had to pick one theory, I would say that...
Morgause is the daughter of Ygrain and Gorlois through a previous relationship.
Morgause is the daughter of Ygrain and Gorlois through an adulterous relationship.
Morgana is Uther's secret love-child.
Morgause is the daughter of Uther and a powerful sorceress.
Arthur and Uther are not related to Morgana and Morgause.
I have a different theory, which I will describe in the comments.
If you have other theories, arguments, objections, or just want to say which ideas carry the most weight with you, please let me know in the comments! (And if you have any impulse to cross-list, I'd love to get a wider sample.)