$50 that they get for free does not have any assigned purpose
That's true. But the next part, that it "can be donated," does not follow. (I am assuming here that by "can be donated" you mean "can be donated without militating against the budget's priorities.")
Consider the following three cases. Case 1: Sheila is struggling to make ends meet. She spends next to nothing on fuzzies or on externality-laden projects (e.g. SIAI). Case 2: Carrie is well-off, and spends a lot of fuzzies and others' utilons. Case 3: Malcolm is in the middle. Sudden income decreases might put him into Sheila's subsistence mode, and sudden increases might significantly increase his fiscal breathing room.
Only in Case 3 should a "free" $50 make a difference in one's spending patterns. Sheila would be better off spending it on food, medicine, or debt reduction; and Carrie is probably already a donor to SIAI (or whatever). However, even for Malcolm, spending the $50 as though there were no alternative use would be imprudent.
The unbudgeted $50 has various possible uses, some of which are better than others. You seem to be saying that a budgeter can, without a utility penalty, spend the new money on __ as though there were no alternative uses to the one suggested in the original post. Is this your intention?
It has to do with mental accounting. For someone who is a) like all humans, a bounded rationalist, and b) like most humans, does not donate nearly enough (from a utilitarian perspective) to charity, a catchy idea to donate unbudgeted money could spur an increase in donation.
In particular, taking a donation out of money you previously thought you would use hurts. Taking a donation out of money you did not know about before does not hurt.
If I gave you $50 you hadn't planned on receiving, would you consider giving it to charity?
Here's your chance to find out.
Just in time for the Tallin-Evans matching fundraiser, ING Direct has started offering a free $50 cash sign-up bonus. I've personally used ING for 10 years and referred over 20 people to similar promotions of theirs in the past so I can confirm that this is legit.1
It's a simple, effective way to get started as an optimal philanthropist for free:
Full disclosure: I was an SIAI Visiting Fellow in 2010. I've also used ING Direct as a customer the past 10 years, but otherwise have no financial interest in them.
[1] This isn't one of those bogus "intro" deals where you have to make sure you cancel the service later on or risk getting charged fees. ING has no fees, no minimum balance requirements, no sleazy marketing emails, and consistently good savings rates. If you want to use them for their good service after signing up, great, if not, no worries. All you have to do to qualify for the $50 is make 3 small purchases you were planning to make anyway with their new debit card. I know it's a trivial inconvenience, but I think it's worth it to be able to donate up to $100 to charity without actually spending any money.
[2] Profit denominated in warm fuzzies, karma, and post-Singularity catgirls.