My D is considering ChemE and Material Science dual major. They are related and only require 14 more credits. She got scholarships and grant that cover pretty much the tuition and can graduate within 4 years due to AP credits. If you have interest in both and it does not cost you extra money or time, why not. It also increase the chance for departmental scholarships. If it will delay your graduation, the extra cost may not worth it. It may be better off to do one semester of co-op.
Hmmmm… DS#1 s degree is Mech E and materials science. Wasn’t a double major or a major/minor. That was a the program/degree. Similarly, DS#2S degree is, IIRC, Chem and biomolecular engineering. Agree that if one can still finish in 4 (or more only if the co-op is included and extends the academic experience) thats ok. But to pay for an extra semester or year to get the minor is probably not worth it. Do it because you want to. Follow your passion.
A minor is a little different situation. Usually, you would use many of your electives to cover your minor, adding little to no time to gradation. In fact, some programs may require a minor (or concentration), as a way to ensure you are picking electives that would add depth or breath to your degree. Your advisor can supply a list of “recommended” minors based on your field of study. Anything from Computer and Information Science to Business Administration to Physics to Public Leadership to Statistics to Sales Engineering to…you get the point. 
When thinking of a minor, balance using your “electives” to fulfill the requirements for the minor, versus using the electives to build more “depth” in your field. As an EE, should you take those extra statistics classes, or more EE electives/projects? You may find the minor more appealing than some of those electives…
I noticed that you’re a resident of California. Lets use the ME program at Cal Poly-SLO as an example:
http://me.calpoly.edu/about/degree-programs/
The ME program offers three concentrations.
•Mechatronics
•HVAC
•General Mechanical Engineering
On the other hand, at UF, a dual AE/ME degree is a popular choice, as is the Combined BS/MS Degree Program (4/1 Program). UF also list several relevant minors, including Biomechanics, Sales Engineering, Biomoecular Engineering, and Electrical Engineering. See the following link for info on these minors (and the dual AE/ME program and combined BS/MS degree program}.
http://www.mae.ufl.edu/current/undergraduate/minors
Where you attend school, will drive what options are available.
On the topic of double majors, the only time I would recommend one is if the two sets of coursework really complement each other in a very meaningful way, but don’t really give you two separate sets of unrelated coursework. ChemE and Biology could be a good one, as could engineering and math, CS with a more hardware-oriented major (probably a science because two engineering majors is really a bad idea, even if you can manage), or something that gives you an understanding of operations/logistics to go along with engineering.
Unless you have a good reason to think that you need a double major, beyond that it simply seems like a good idea, I really don’t recommend it.
At my son’s school, an AE minor only requires two more classes with his ME major. So even though he is interested in AE as a career, he picked a broader E major with a more focused E minor to keep (or give him a sense of keeping) all career options open. Although I understand that, per the sage advice of many on this board, including boneh3ad, AE employers hire ME majors, and ME employers hire AE majors. So it may not make a difference. But he never considered a double E major.