I'd write her a thank you note and include a $25 gift card to a local shop or restaurant.
This. I wouldn't exchange any money, but a gift to thank her for her consideration is nice. Or if you're friendly enough, treat her to lunch some time.
I don't get why on EARTH she would choose that time of day (unless guaranteed overcast/cloudy) for a photo session. She should know better. Of course, $125 IS super cheap so... I like the gift card idea.
While it'd be generous of you to pay her. If she botched the session by having it at a crappy time of day and not consider her surroundings and is saying it's a freebie I'd take it. It's a lesson I hope she learns in time. Even if it looked cloudy a photographer should try to hit golden hour which will regardless of the surroundings at least ensure softer tones on the skin and an easier time with the eyes in the sunlight.
When I was starting out this was a huge reason I didn't charge at first I needed to get to a point where my good to bad shots ratio was acceptable. If I was putting out close to no good shots then it's was a freebie and likely ask for a free make up so I could improve.
I'd always appreciate the client saying "are you sure you don't want me to pay you?" Even though I'd decline it was a nice gesture all the same. I would likely offer a follow up session for free as well.
I think it's really nice of you to consider paying her. Everyone is different though so please take what I say with a grain of salt.
I also have to make mention that I feel this and overall fauxtographers cause clients to wonder why photographers charge so much for the sessions when {insert persons name} can do it for far less. That's when I tell them to go to them then, often they come back after a not so great session or horrible customer service. (not all the time mind you but some)
Sorry off topic OP. The gift card was very very nice of you!