Well, the most common answer - so common it's a meme - is play what you like. Just about any game can be the basis of a good speedrun if it's something you're willing to play long past its replay value as a casual game. Try to narrow the umbrella a little bit - what kinds of games do you enjoy? Platformers? RPGs? Shooters? What about speedrunning draws you in and makes it exciting for you? Super-advanced movement tech? Route planning and creative use of items? Glitch abuse? Your answers to those questions can point you in the right direction. (I like RPGs and I like abusing glitches. Pokemon Gen I. Or, I like platformers and abusing glitches. Donkey Kong 64.)
When you say "web," does that mean it has to be browser-based? If not, not to be self-serving, but I just started a speedrun.com hub for a nifty series of puzzle platformers by NAL Games called Innoquous. The first four are freeware, available for download from his site.
Edit: I should have checked your profile first. I'm sorry, didn't realize you already ran tons of games. The questions are still good to walk through to point you to your next conquest!
http://www.addictinggames.com/puzzle-games/redremover.jsp
Who hasn't played this Red Remover in their elementary school computer class? I used to play this game (amongst others like Red Beard and Red Ball) and Mr. Chavez would get absolutely FURIOUS with me and would sometimes send me outside the computer lab where I wasn't back in the rest of the period. A lot of the times I really let his yelling get to me personally, but that's just how 8 year olds function.
Haha, but that's besides the point. In all seriousness, Red Remover ( http://www.speedrun.com/rr ) is an outrageous speedrun using crazy strats like clicking on the squares as fast as possible, navigating through the tedious menu without your mouse freezing up ( I should really consider getting a new mouse since mine has been acting up like hell, but what can you do when your nearest electronics store is a 45 minute drive without traffic), as well as making sure you time your runs properly.
It's a great speedgame for beginners because there is no RNG, and it's a very quick speedrun (6 minutes if you're good like me; but can be about 6 mins and 54 sec if you're not that good with your hands or using your brain for optimal hand-eye coordination), and it brings back lots of nostalgia like Mr> Chavez kicking me out of the computer lab.
Overall, I would recommend Red Remover (and Red Beard) to anyone
There is currently a LIVE bounty on Bloons Insanity as well.