There are actually two different questions hiding here.
Why is the electron negative?
That part is convention. Benjamin Franklin could have named the electron's kind of charge “positive” and the proton's “negative.” Nothing in physics would have changed.
But: Why does the electron have electric charge at all?
That's a much deeper question.
The electron is an elementary particle, and its electromagnetic charge is built into how it participates in the Standard Model's gauge symmetry. Physics tells us remarkably precisely how that charge behaves, but doesn't currently derive the electron's charge from something more fundamental.
So the minus sign is our label.
The charge itself is not.
