As a Lovecraft fan, I think that his work would make for a good game, if applied appropriately. But as Yahtzee points out, not every Lovecraft story or theme makes for a good plot or game mechanic, respectively.
Sanity is definitely one of those. If the player character is going to be going insane, it should be brought across through things like dialogue, sound and graphics (that is clearly-visible images and not just acting like someone's playing with the focus on the in-game camera) but not an in-game mechanic like a meter. Because honestly, where exactly has medical science found a way to measure sanity in a quantifiable way?
Now, on the idea of Lovecraft stories as game plots; again, Yahtzee's right in that some of Lovecraft's ideas of "horror" don't age as well as others, so the horror element is rather lost. Though I don't agree with him on the idea of everybody's accepted we were some kind of evolutionary accident in a godless vacuum of uncaring void. If I could be allowed to speak as a religious person for a moment, I think a lot of people have accepted the idea of life outside of our own planet but realize that doesn't necessarily mean that were entirely unimportant, nor does it disprove the existence of God, even if it turns out that from a physical or intellectual standpoint we're not the pinnacle of all life in existence.
Right, that out of the way, let's get back to Lovecraft as a game writer. A lot of the reason Yahtzee gives for Lovecraft not translating well into a game idea seems to hinge on the fact that not everything he wrote is that horrifying anymore. It's true that "Shadows out of Time" wouldn't make for a very scary story; the aliens are in fact very courteous (if you can forgive the whole hijacking-your-body-to-take-a-vacation-on-earth thing). Even "The Whisperer in Darkness" wouldn't be that unsettling, if the aliens in that one were, in fact, willing to keep your body on ice while taking you for an interstellar road trip. Although I have to say if one of the protagonists was in the right of it, it sounded like the aliens scoop out your brain and take you with them whether you like it or not, which could be rather unsettling.
But there are Lovecraft stories whose elements of horror last to this day. Take "The Rats In The Walls". I think we'd all feel a little queasy if we found out our ancestors were the wardens of some underground society that were cannibals eating the flesh of primitive cavemen. "The Case of Charles Dexter Ward" or "The Dunwitch Horror" are other good examples, where delving into unhealthy and sinister mystic practices unleashed things that could threaten the world.
Now I can see an argument for how this might make good movie material, but not game material, and you'd be partly right. An existing Lovecraft story might not make a good game in itself, because when you read Lovecraft with modern sensibilities and mindsets, you realize the protagonists in his stories were often trying to delude themselves or were incredibly naive, narrow-minded and dense to blunder into the situations they did. The only way to get the player to follow in their footsteps is to box them into it.
But Lovecraft created a setting for his stories that has a lot of potential: the early 20th century, when industry and science are growing but there's still people and things around us that defy scientific explanation or explanation through accepted preconceptions. It's a setting where a police investigation can lead to a crazed cult whose ringleader is attempting to find a way to unnaturally prolong his life, where a scientific expedition can discover a lost, ancient city ruled by creatures that weren't human. Even after all these years, the world Lovecraft created with his stories remains fertile ground for inspiration.