I don't think it's the only factor, but it certainly can contribute. I remember learning in intro psych that there's a certain balance of stress (eustress and distress) that is optimal for happiness-- we don't want too much but we do want some. That idea that idleness is not the key to happiness has definitely been a factor in a lot of literature, and is the moral of Candide-- To find happiness and contentment, plant your own garden.
Also the Whitehall studies of British Civil Servants found that people are happier and have better health when they feel as if they are in control of their environments. So I'm not sure that it's just busy-work that's needed to fill the space, but something that helps you feel like you are changing your environment for the better.
Myself, I definitely feel happier when I'm productive-- when I actually produce things that I can be proud of or when people say I've been helpful. My research may be going badly, but if I can say, "I made a great dinner" or "I helped a student be less afraid of polynomials" ... it gives a little boost, at least I'm good at something. One of the things that helped me when I had really bad anxiety-induced depression was starting to do RA work...positive comments from my supervisor gave my therapist ammunition to shut up the negative repeating voices in my head. They could tell me I was stupid, and I could counter with, My advisor said I did a good job finding that mistake in the paper and running regressions to fix it. | 
