Wikipedia is so unreliable. I personally always thought it was because of Kenny Stabler and/or Jake being elusive.
Here's a piece from an article in an Idaho newspaper:
“Jake the Snake” Plummer got his nickname in junior high after he read the autobiography of another quarterback known as “Snake,” Raiders great Ken Stabler. The moniker fit the skinny Plummer, who was known for his ability to elude defenders and make plays — leading Capital to a state title his junior year and a runner-up finish his senior year.
I'm guessing it's a mix. I'd wager if his name was Thomas it wouldn't have stuck.