Hopefully I can be helpful in answering this one - even though the majority of Investment Bankers tend to major in Finance or Business/Economics, there are also a good amount of bankers who have backgrounds ranging from Technology/Engineering (such as myself) to History/Arts.
I don't believe there is one combination or one major that will best position you for a career in Investment Banking. As long as you have good reasons for why IB makes sense given your background, anything is possible. There are, however, best practices (in my mind) to position yourself for recruitment come Junior/Senior year:
1) Have an interest in Finance, and specifically in Valuation 2) Find internships in the field of Finance to exemplify your interest during your Freshman/Sophomore/Junior years 3) If you didn't major in a business-related field, have a good reason for pursuing a career path in IB
Note that in IB, there are multiple verticals/industry groups - you can pitch yourself as interested in a specific industry group given your academic background helps you better understand that industry. For example, if you chose to Major in Mech. Engineering, you could say that your academic background better positions you for the Industrial industry groups when it comes time for recruiting.
If your end-game goal is focused on IB/Finance and nothing else, it will likely be best to major in a Finance related field. However, it is still possible to major in Mech. Engineering (if you want to consider jobs in Engineering) and recruit for IB if you find that those jobs weren't a good fit and IB is what you want long-term.
Regarding Minors/Majors, it might be good to balance a quantitative field such as Financial Engineering, with a minor in business (for soft skills, communication, or to learn more holistically about business) or something strategy/macro related (economics, etc.).
There are a ton of materials out there to help with IB recruitment (WallStreetOasis / Megers&Inquistions / Vault career guides), so definitely try to utilize those as well as any alums from your school to help with this pursuit. Hope this helps!