Idea #2 - Sandbox, Stealth, Thief, Zombie game
You play a master con artist who moves into a medium sized town to begin another double (triple? quadruple?) life to con more people for more money. He has a HORRIBLE nightmare his first night staying at his new place. He dreams that some sort of apocalypse happens in the town, where houses are burning down, car are crashing into things and immediately exploding, people are dropping dead in the streets, and a few other specific things happen while he runs around outside in a panic. One of those is a disembodied, decayed head floats in the middle of his range of view, laughing maniacally, floating just barely out of reach, but no matter where he looks, the head stays there, making the player wonder if it's an illusion or something else. Then a gigantic dagger falls from the sky, stabs his house, then the town is consumed in a firey explosion. Then he wakes up, freaks out a little, calms down, and goes downstairs to get some water to calm his nerves. Everything outside looks normal, although really... really quiet. Then one way or another, he ends up in a locally owned store 'cause he's robbing it 'cause he can't find anyone around. Then he realizes someone is in the store and the person is a zombie. The zombie tries to kill him, but ultimately the con artist finds an axe and chops the guy's head off. Then the con artist realizes that the head he just chopped off is the same one that he saw in his dream. On top of that, more zombies are coming 'cause they heard the noise.
The con artist has to learn how to live in his new zombie infested town by getting food and water, all the while stealing money and expensive stuff to pawn once he finds a way to get out of there. He boards up the windows and doors of his house (except the window on the top floor, which he uses to get to the rooftops of other houses) and then only goes out at night. The point of this game is keeping quiet, sneaking around, and steathily killing zombies in quietly gruesome ways to avoid attracting attention. Dunno what incentive we could give for collecting money, powerups probably. But to keep the game from getting too repetitive, we set up points in the game where parts of the nightmare come true, semi-constantly trying to freak out the player. These events would always occur in order, no matter how many times the game is replayed. But since this is a sandbox after all, these events could occur in many different places, perhaps even some of the same places as others. That would make sure that players don't need to wander far or explore everywhere to advance the story. The con artist would keep a journal of the events (players can read his entries too), his money, supplies, and also which houses he's hit and which ones he's sure don't have anything of worth to make things easier for the player. The money status of those houses would be a display option on the map.
Now for the various ways you get to kill zombies

The simple axe to the head is effective enough, but there are more ways. Raid a gun shop for silenced or not silenced weaponry (if you're looking for attention). Collect a bunch of silver items, melt 'em into sterling silver, and use that as a much more effective melee weapon. We make sure the players know that making a silver weapon is a possibility. You also find grenades (where though?) and you can make molotov cocktails. Noise is also a weapon. Set a car's alarm to go off, climb up to a roof, board up the doors and windows, and set off the alarm. Watch the zombies swarm, then throw a molotov from the shadows. Zombies trapped in a house that doesn't seem to have much worth in it? Burn the house down. I guess we could have a vehicle ability too (I could easily see it attracting zombies), but would probably be best used along with a noisy distraction, but would be a pretty nice way to travel fast.
End of the story: the con artist sacks a museum, and discovers that the source of all the zombification of the town is from a ancient bone dagger in the local museum. It looks exactly like the dagger in his dream. He steals it, reads up on it, and finds out that it's rumored to do quite a few things. It's rumored to be a way for someone to achieve immortality. It turns everyone within a certain radius into zombies when its "chosen wielder" stays within that radius for one day (it's not a constant effect though, he can take the dagger out of the radius and not make more zombies). It's also rumored to kill zombies on contact, no matter how trivial the cut, and also give the zombie's soul to the wielder to allow them to live longer. What that means for the player is that this weapon is god mode. Each zombie kill restores a lot of health, and any hit from the dagger kills a zombie. You'll definently want to make a lot of noise just to kill a bunch of them. The point of killing zombies with this weapon is to collect souls. After hitting 1000 souls, the wielder becomes immortal, at least, that's what his research says. The research also says that the people can be changed back to normal, it only requires that a currently living person be sacrificed using the dagger. The chosen wielder can sacrifice himself if needed.
And that's as far as I've gotten. I'm not sure how the ending would play out, or how the dagger should really work :S But I think the main strengths of this game will be atmosphere, the game is meant to creep you out.