In The Amazing Spider-man, Peter Parker is poking around Oscorp when he goes into a silk thread producing chamber. He disrupts the machine (or something) and all of the spiders fall to the ground. One of them ultimately bites him.
Was there something special about Peter, or that specific spider, or could anyone gain the same powers by being bitten by one of those hundreds of special spiders?
Updated based on The Amazing Spider-Man 2
No. The spiders would not have successfully given powers to just anyone.
In a video log, Richard Parker explains that his own DNA had been fused with that of the spiders. Only someone with shared DNA (such as Peter, or likely Ben Parker) would successfully gain powers from the spiders.
Original answer based on The Amazing Spider-Man
In-Movie answer: No, not just anyone would probably gain the powers.
Dr. Connors mentioned that every cross-species DNA merge resulted in death. Even with what should have been a stable formula, Connors body did not react to the formula the same way - changing him into a giant Lizard while Peter simply gained powers.
Not-Quite-In-Movie answer: There is probably something special about Peter.
There is a fantastic write-up about The Untold Story which was cut out of the movie. It discusses hints of deleted scenes that were in trailers and pieces them together with the final film to theorize the supposed "Untold Story".
Months ago I told you guys that I heard rumors that The Amazing Spider-Man would be making a simple, but huge, change to Spider-Man’s origin. No longer would the spider bite change Peter Parker into a superhero. Rather, the spider bite would activate something already within him that would make him a superhero.
(snip)
The first major hint is still in the movie. Curt Connors is talking about how every other subject upon whom cross-species DNA merging was attempted died. He does not know that he is speaking to the one success story. But how did Peter survive? The movie leaves this sort of dangling there, but the clues are in front of your face. Peter was bitten by a spider... a spider that Peter’s father bred. A spider like the one under glass in the film’s prologue. A spider like the one on the chalk board in his father’s office.
(snip)
In the first and last trailers we hear a man - clearly Mr. Ratha - whispering ‘Do you think what happened to you, Peter, was an accident? Do you have any idea what you really are?’ That certainly sounds like a reference to Peter’s genetic destiny, and a clip that I don’t believe is in the finished film. Judging by the whisper I’m going to guess - and this is just a guess - that it is Ratha’s dying words to Peter after the Lizard does him in. This is a movie that seems like it should have at least one info dump dying declaration in it, and this would have been it.
This is a really late answer, but the reason Peter actually got his powers was through ONE spider being zapped by a powersource (presumably radioactive) and falling on him. The other spiders were totally normal, it was only that one spider which bit him. It's easy to miss, but after Peter messes around with one of the "webs" there is a shot of the spider being hit by the bolt of energy.
So Peter is a freak accident, unrelated to the process Connors used.
