Sequels for me are a tricky situation. In many cases I feel they're unnecessary. To me an ideal sequel should include the following:
The sequel should have ties to the story of the original game: Be that continuing a story that was unresolved in some way, or a setting up a new story for the same characters.
If there is a new story, set it up in a way which is actually interesting and don't recycle story concepts from the original.
If there are new characters include some older familiar characters so as not to alienate the core fanbase.
If there are characters from the previous game, have them advance as a character. None of this nonsense where they're the same in every game.
If their character does advance in one game from the previous one, make it something more interesting than the standard: 'Now I'm jaded and don't want to be a hero' nonsense we get so often.
The gameplay should improve upon concepts established in the previous titled, refining what was good and removing or smoothing over what was bad. It should also introduce other, new, gameplay elements so as to keep things interesting.
The game should be accessible to new audiences so as not to alienate rest of the user base.
If there is multiplayer it should do a similar refining to the single player, adding in a mix of new and old maps and adding new elements and game modes.
The sequel should stand on it's own as a game.
The sequel should not take away support and DLC for the previous title in the case of certain games, obvious example aside.
The seqeul should not make itself stand out as a new game by introducing a gimmicky gameplay element or character.
EDIT: The sequel should be well-named. A simple number on the end is not a compelling reason for me to buy the new game.
If a sequel manages to fulfill those criteria, you have yourself a solid sequel.
EDIT 2: For sake of example, let's take how I would like Mass Effect 2 to improve, let's put aside what has been revealed about the game so that I can fully take advantage of the list:
I would like to be able to switch between squad members to be able to give better combat directions, and just generally enjoy that character more.
I would like more interaction between Shepard and the squad members.
I would like to have the combat made more dynamic with a wider variety of combat situations.
I would like more customization in terms of cosmetic elements.
I would like to have appearances from many characters in the first game.
The original set up for a larger story, I would like to see that story advanced on.
I would like to see the characters from the first game mature and advance as characters.
I would like to see new characters fill more of a role than just what they can do in combat.
I would like to see some more interesting and varied side quests.
I would like a longer main story arc.
I would like more varied experience of the game based on playstyle.
I would like fewer driving sequences.