This is all posted assuming this is a real story, which it isn't. That said,
1) It's a stack of 20s. I don't always pay my rent that way, but when I do, I'm a drug dealer.
2) Someone found my $1000 worth of $20s. If I'm a drug dealer, that person isn't returning the money. If I'm a dude, that person is going to be less likely to meet me to give back the money, probably be cause he's afraid I am a drug dealer, or thinks I will be angry because the finder "stole" the money, etc. If I'm a scatterbrained woman who offers a sob story, and I offer a reward, I might be able to get a lead on my lost money from a white knight, Dudley Doright type person. The alternatives are a) no one found the money and it is gone forever, b) someone found the money is keeping it, or c) someone found the money and wants to return it. As someone who wants to find your money, your only option is to appeal to the 'c' group, and assuming you can contact the person who found the money, you must present the most appealing situation possible, which is helping a "damsel in distress". In case the finder of the money is a woman, there should also be an incentive, hence the reward.
So, precisely because it is a "damsel in distress" I am immediately more suspicious than I would otherwise be. The reward for returning cash always strikes me as odd, because if the finder is motivated by money, you've got a big problem that a reward can't solve. Like you pointed out, a person motivated by honor doesn't require or even want a reward, and a person motivated by a reward wants money they already have.
3) Just because "she" provided info that only she would know, doesn't change the fact she could be a fictional character made up by the person who lost the money. She could further be represented by a real-life female who is associated with the person who actually lost the money.
I've had sketchy people come after me with baseball bats for giving them dirty looks. There's really no barrier to low for these types of people that will cause homicidal rage. Taking $1000 of their cash, even if you are in the process of trying to return it, will not be seen by those types of people as a good deed. They will try to get their money back, and beat (or kill) the living hell out of whoever "took" it.
I'm just saying, for me, in this fictional scenario, it doesn't pass the sniff test.