Can a tweet be added to graduate research proposal? Can I add a tweet by someone famous in the field into my research proposal to show why is it important? 

I am asking this because I have not seen anything like this but people do use quotes by famous people in their thesis. 

If the answer is yes, then how will I be citing it?
You can quote any source, including Twitter. Cite it in the same way you'd cite any web source and be aware that it might disappear.