One factor that would convince me to do a job for less money would be for a family member or family friend. Since the fees and expenses of an invoice are negotiable, I can give a break to this individual and charge them the hardest bottom line that I can. Depending upon the relationship I have with this individual and if this job will bring me future jobs, will determine the level of cut backs I can make.
Pro-bono work is a job done completely for free and usually done for non-profit organizations. I do see the need for these kinds of jobs. I also think it can create a volume of experience for everyone involved. Helping the world around you can only keep your career rolling an easy downhill trek instead of uphill journey.
Good repeat customers are a great reason to think about giving a break, just once or twice. You do not want all your customers thinking you can give them the same break to everyone that you gave to one singular situation. However, a good relationship creates the best word of mouth advertisement and thank heavens, THAT you can afford.
What would not make sense to charge little or no money:
I do not believe first time customers should get a major deal of fees and expenses. It is different if there is a discount or a special on studio time/ assignment particulars because the client will know they are getting a one time deal. It shows that you value yourself enough to charge