Validity of conditional statement when the premise is false.

Well, a big reason that we chose this convention is that we are sort of short on options:

• We could say that a false premise implies that the implication is true exactly when the conclusion is true, but that would be odd because then the premise doesn't do anything.
• We could say that a false premise implies the implication is true exactly when the conclusion is false, but eww.
• We could say that a false premise implies the implication is always true, which is what most people who think about the alternatives do.
• We could say that a false premise implies the implication is always false, but if we do this then $p\to q$ has the same truth table as $p\wedge q$, which isn't bad but it seems like there is something more that we want out of an implication than a simple 'and' statement.
• We could reject the law of the excluded middle so that the implication is neither true nor false. This turns out to be a valid option, but also eww. More practical reasons to dislike this is that it destroys double-negation $ (\sim\sim\! p$ is $p)$ and therefore contrapositive and contradiction proofs.

So you have to pick one. They're a sorry lot, I admit, but we're stuck with them, and one has proven to be more realistic and pragmatic than the others.