Easy: Azathoth, Yog-Sothoth and Hastur
Medium: Yig and Nyarlathotep
Hard: Shub-Niggurath, Ithaqua and Cthulhu
Regardless of the Old One, once they've awakened they're very difficult to beat. Winning that way will always be tough! For that reason, I consider Azathoth to be one of the easiest to defeat. Sure, when he awakens,
Azathoth destroys the world (which kind of sucks), but that's his only ability and with a huge 12 Doom required, the investigators have plenty of time to get their affairs in order and start going after Elder Signs.
Yog-Sothoth is also not too tough. Despite having a
[do X] or be devoured attack, his ability (which is an alternative awakening trigger) is related to the Other World cards. The Other World cards typically have good rewards, but you can definitely win the game without them, so it's not such an issue to just ignore them.
Next is Hastur, who also has an Other World-related ability; his adds Doom. The obvious strategy there is: stay away from Other World cards. He has a brutal attack which could lose 2 Sanity with each attack once he's awakened. Without clue tokens to defend with, most players won't last more than two rounds. What's more, his task can be very difficult since it gets tougher when there are more monsters on the board (and he generates five himself). If he awakens, he's very tough to defeat, but, like Azathoth, he has a long Doom track making him not such a threat.
Yig and Nyarlathotep are more intermediate. Both have short Doom tracks (8 and 9 respectively) so you don't really have much prep. time before you have to start going after the Elder Signs. Also, they both have strong monster-synthesis. Yig pulls out five monsters. With only 8 Doom, that means they come out fast. Nyarlathotep's ability adds the incredibly tough Mask Monsters to the pool. They keep coming back and don't give out trophies and when they land in the wrong place can cause a real headache. When these two awaken, they're not too tough to reduce Doom for, but Yig devours any player that doesn't lose a Common item, a Unique item AND a Spell (a high price if they can even afford it!) when he awakens and Nyarlathotep's attack can devour players.
Shub-Niggurath is the toughest monster-related Old One. He pulls out seven monsters AND adds to the tasks for all of them. This can make some of them incredibly tough to defeat without items. Once he's awakened, his attacks can devour too, unless the players have trophies. Fortunately, with the number of monsters around, there should be plenty of trophies available by the time he comes out!
Ithaqua is very tough. His ability punishes any player who uses a Unique item or a Spell. It's almost impossible to play the game ignoring those two, so this will definitely sting you at some point. Over the course of the game, this can sap you quite a lot, and make players need to spend turns to heal up rather than completing tasks. When he awakens, there's a 50% chance that each item will be lost, and when he attacks he reduces two Stamina at a time. This focus on just one of the two health stats makes him especially tough for the low-Stamina characters.
And, appropriately enough, I consider Cthulhu to be one of the toughest. It's a toss-up whether he's tougher than Ithaqua and/or Shub-Niggurath, but I'll list him last because he's earned the prestige (:P). Straight out of the gate, he reduces everyone's maximum Stamina and Sanity by one. He's incredibly tough to defeat if he awakens, since he heals during every attack and slowly does unhealable (and unpreventable) damage to all the players. Fortunately, he has a long Doom track (11) and does damage very slowly when awoken, but the reduced maximum to the two health stats is such a significant, unignorable disadvantage that it makes him much tougher than others with long Doom tracks like Azathoth and Hastur.