Both work

I agree with Glenn's answer ("stand by" as per the dictionary definitions and "stand with" being commonly used). And I would add that the examples of "stand with" seem to be most often used in situations where there is actual conflict. There is a common idiom "to stand shoulder to shoulder with" that is often used in this type of situation. I would speculate (I have no evidence for this) that it might be a really old idea that goes back to ancient times when soldiers had to stand next to each other to form a defensive wall, and then gradually over time people forget the original meaning and it gets mixed up with other ideas (like 'to stand by')...so in other words they are both correct.

In every day speech they mean the same thing

They are equally correct. Use the preposition whose meaning agrees with what you need to say. “By” means “next to”. “With” means you will remain together. It does not matter at all which one people use more often.