Race has nothing to do with what spiderman stands for, so yes, I could see a geeky black, asian, or hispanic kid doing a damn fine job portraying the iconic do gooder.
Oh come on, dude. Get real.
It's part of who the character is. That's not to say you couldn't have an equally effective super-powered do-gooder of a different race, but Spiderman -- as in Peter Parker -- is white, just like John Shaft is black, just like Liu Kang is Chinese, just like Katana, Doctor Light, and Silver Samurai are Japanese, and so on and so forth. It's just part of who they are.
Now, sometimes you can mess around with this stuff and it isn't a big deal, but I think it really depends on the character. Also, the notion of "Let's just make Peter Parker black with no real explanation" is far different from, say, creating an entirely different character like the new Mister Terrific or John Stewart -- one of the Green Lanterns -- are both black. But those are different characters entirely. Different people donning the mantle of a given hero. Like everyone knows that Batman is Bruce Wayne (white, by the way), but for a brief period in the early 90s, some french (or Quebecois? I forget) guy who went by the name Azrael ended up taking over the role. I heard Dick Grayson recently donned the cowl too, but all that is while Batman is supposedly dead (right. I'm so sure).
Regardless, the point is not "what they stand for", but rather who the character is. The further you get from that, the less you have to do with the character and the more you might as well just do a new character. There's nothing stopping Marvel from creating ANOTHER Spider-man. I mean, they've done it before with the Ben Parker clone. But that's a separate and distinct character from Peter Parker. Peter Parker is Peter Parker in all his whitebread glory, just like Superman is not an Aztec (never mind what you saw in Infinite Crisis...). Unless you can provide some in-universe explanation for the change (IE: Psylocke became asian after she was killed an reincarnated by the Hand into an asian body), don't mess with the hero's identity.