From a financial standpoint, how much more can I expect to earn after my degree?

Pursue the dual major because it interests you and this is your opportunity to do it, not because you expect a higher salary, because the higher salary probably won't happen.

I spent a long career in industry in engineering before coming out of retirement to teach a few years ago. In my experience, hiring managers generally don't care that someone was a dual major. Part of this may be a reflection of the fact that managers tend to hire people who "look like them". Not very many managers were dual majors, so they don't value it. But there's a practicality to it as well: If they're hiring you to be an engineer, it's probably because you are an engineer, not because you're also a mathematician.

What hiring managers do value is the master's degree. Again, this may be in part because so many hiring managers have master's degrees (which many of them may have earned part-time while working). Hiring managers tend to have master's degrees, so they like to hire people with master's degrees, and they will pay more to get them.

If your concern is best financial return on any additional effort beyond what it takes to get a bachelor's degree, I recommend going for the master's over the dual major. Many universities have programs that allow students to go straight through to their master's, double-counting some of their credits and earning both degrees with just one more year of coursework only, no thesis required. For example, here at University of Michigan in computer science, where I teach, it's called SUGS (Sequential Undergraduate/Graduate Studies); to be admitted to the program you need to be a junior or a senior with a 3.6 GPA. Your school probably has a similar program.

If your objective is a career in industry and you're concerned about "bang for the buck", the master's is the way to go.

Expecting a difference in salary is not reasonable. It just wont be that simple. Your expectations will not be falsifiable.

You can expect to have more options. That is where the value here lies. You will definitely have more options to choose from, one of those options could include a higher salary, but also working environment, residency location, etc.

Having more options is always desirable.

If you are interested in mathematics, but are also trying to maximise your future earnings, I would recommend a minor in math, then getting an MS in mechanical engineering.

A dual major won't earn you any more on its own. It might open additional opportunities, but since an undergraduate degree in pure mathematics likely will present less opportunities than an undergraduate degree in mech engineering, the benefit there is small. (but those additional math opportunities are something you might want to keep in mind if that is what you want to pursue, but at that point, why get a degree in mech eng?)

A masters degree will earn you more in engineering though. I will give you an example. The first company I worked for, a big bay area tech company that everyone has heard of, had this as their hiring policy: $x for new grads with a BS, $x+10k for new grads with an MS.

A minor in math would look good an a graduate application. A graduate degree in ME should earn you more money. So that path seems to check all the boxes you are looking for.