If that were true, I would expect that depression would make you creative, so you could think of creative solutions to your problems. My experience suggests the opposite of this post... when I had a big problem in my life and became depressed about it, I believe that depression made solving the problem significantly harder. In order to solve my problem, I needed to try many different things, none of which had a high probability of working. Depression made me pessimistic that any of them would work, so I was not that motivated.
This bothers me too. In my own experiences with depression, it certainly led to rumination, but not particularly useful rumination. Mostly the same thoughts over and over, and mostly unhelpful ones to boot. (Edit: typo.)
Since there are intelligent people here who follow the topic of evolutionary psychology, I'd like to hear opinions about some research from 2009. Particularly if this idea seems reasonable or not, but possibly other opinions that people might have about it.
The idea is a variation on one that's somewhat popular here: that some conditions usually regarded as mental illnesses (Asperger's for example) are beneficial, even adaptive. But the condition in question now is depression. Briefly, the argument is that depression, at least when it is a response to stimuli and not a permanent feature, can have the useful effect of encouraging more rational thought when this is particularly important, even at the cost of quality of life, and that this is adaptive.
Links: a Scientific American article, a journal article (which I haven't read, behind a $12 paywall). Here's the abstract of the journal article:
The full journal citation is Andrews, Paul W., and Thomson Jr., J. Anderson; July 2009; Psychological Review 116 (3), 620–654; doi 10.1037/a0016242.