Hi MAH,
Thanks for your message and for giving me the opportunity to learn about your personal, academic and professional background. After reviewing all of the details in your profile, your undergrad institution is not a concern to me since there are many international students accepted to HBS who have unique undergrad institutions which they have attended. I know this from personal experience as I have worked with candidates who were accepted to HBS and fall in this category.
As for an MSc, I would not recommend doing a MSc to offset your undergrad institution name. This would not help your candidacy and may be a waste of time. I would spend this time on gaining additional extracurricular activities or work experience in the tech industry or with a start-up to gain the entrepreneurship (non-family business) experience. I would also try to diversify your undergrad majors and not choose Finance and Accounting as you suggested.
Lastly, since you have not taken the GMAT exam and your graduating GPA is unknown at this time, it is difficult to say your chances of getting into business school. I think you have conducted some great introspection and I suggest that you continue to do so moving forward. There are many things working in your favor and the best thing that you are doing is starting early!
Thanks for contacting us and I wish you the best in your MBA journey.
Best regards,
Sue