I would leave things alone while doing more homework first.
My first guess is that the puddle does not pose a threat to you, unless there are unusual hidden underground features such as gaps and interstices between underground rocks, or unusually porous soil.
If the water is sitting on your neighbor's property I would not dump dirt on it. Your yard might be the second lowest spot and dumping dirt on your neighbor's low spot might make all of the water come onto your property.
You can quietly dig a trench for a French drain almost on the property line and on your side provided that the bottom of the trench goes downhill towards the storm drain or wherever you want the water to go.
You do not need to put gravel in a swale unless you need the swale bottom to go lower to achieve the desired downhill slope. The gravel keeps people from falling into a swale parts of which have deeply sloped sides (the degenerate form of a French drain).
Thank you - I've decided to just leave things alone for the reasons you noted above. Our soil is very clay and water is actually contained within a low spot on their property over 20ft away. In fact, my property line appears to be about 6-8" higher than their low spot and I never see water entering my property, at least not yet. I do have a storm drain on my side yard just in case it does.
I thought about creating a little swale but I would have to cross in to their property to divert water down.