OK, I'll take the contrarian position here.  Someone has to, anyway!

Arkham Horror:  Simply put, it's Clue:  The Cthulhu Edition.  It's as if someone started out with Clue, added a Lovecraftian theme, then added in a whole lot of bits and pieces just because they could, then went to FFG with it,  and they said, "Looks good, but needs more pieces."  I played a couple of times, I don't recall finishing a game, I only have vague memories of it becoming clear that I was going to lose because I didn't have enough moves to get from the Boarding House to the Kitchen to retrieve either the Elder Sign or the Candlestick, I can't remember.

The theme is strong, but underneath, it is, in my opinion, an overly complicated and overly random board game.

Mansions of Madness:  Honestly, haven't played it.  I looked at a video demonstration and it struck me as having a few more RPG elements than Arkham Horror, but other than that, being another rehash of Clue.  Only literally set in a mansion, and with many of the same rooms.  I can't verify whether it has a Candlestick or a Lead Pipe, but I wouldn't be surprised.

Elder Sign:  It's Yahtzee:  The Cthulhu Edition.  Or as one wag on BoardGameGeek put it, Yahtzee-Sothoth.  Again, the theme is strong, but underneath, it's a dice-rolling game.  Its main selling point is that it's fast, and so for that reason I do play it on an occasional basis.

Call of Cthulhu LCG:  OK, I'm tempted to say that it's Magic:  The Cthulhu Edition, but it's not quite that.  The gameplay is definitely different enough that the similarities end with the fact that it's a customizable card game.  The theme in the LCG is very strong, and the gameplay is cerebral and challenging, as befits a Lovecraftian themed game.  Its main weakness on the theme front is that players do play both sides - not just the investigators, but quite frequently the monsters, and very often a strange mix of them.  I don't have as much of a problem with that, but for some people it does ruin the mood.

Call of Cthulhu Role-playing Game:  For theme, the RPG is the best, hands down.  It does require a very good GM who can craft epic Lovecraftian tales, and without that runs the risk of devolving into really bad schlock (although I am sure there are players who play it for that, and love it for that).  If deep immersion in the Lovecraft world is what you're looking for, it's all about the RPG.  The main reason it's not as popular as the other titles above is, well, it's an RPG, and requires a bigger commitment of time and learning the rules, as well as interaction, than a board or dice game that's straightforward to learn and play.

In my opinion, "best" depends on what you're looking for.  If it's theme, the RPG.  If it's challenge, the LCG.  If it's accessibility for a broad audience, Arkham Horror.  If it's cool looking components, Mansions of Madness.  If it's fast play with easy rules, Elder Sign.

I largely agree with Runix.  If I want a cooperative game rich in the lore I play the RPG.  If I want a fun competitive game set in the universe it's the LCG hands down.  I've only played Arkham Horror, but it takes as long as an RPG session, and has all the limitations of a board game.

Runix said:

To win a game of Arkham Horror all you need to do is to close the gates that keep popping up. It isn't about collecting clues (or are you referring to clue markers?). Anyway, I enjoy AH quite a bit and that's mostly because it supports solo play - the only other game that works (well) solo is Elder Sign.

Of the 200+ games of Arkham Horror I played only about 10% have been with multiple players. Fortunately, the long play time doesn't matter in a solo game, since you can just leave the board in place and continue at a later time.

I've recently read the basic rules of 'The Trail of Cthulhu' RPG which uses the GumShoe system. It definitely made me want to give it a try. I feel the rules are a better match for the typical Mythos game play. Has anyone here tried it?