A French drain is one way to solve the problem, however I think you would lose that tree in the process of installing it. It's about as simple as you say but you should be back filling with river rock or larger gravel. The pipe needs holes. The trench should be at least a few feet deep and wide.

Working around those big roots could do a lot of damage, possibly killing the tree. Then you've got another problem on your hands. This is the most expensive and effective solution.

Another solution would be planting plants that like lots of water or possibly another tree. You would lose that area as functioning space but you'd also have some nice foliage to glance at. Kinda like a little wild garden.

The water has to have somewhere to go. Actually you should go about 8 inches deep, lay the trench with a landscaping fabric, lay the pipe on top of the fabric and cover with a clean washed gravel.

Then wrap the fabric over top of it. Then put dirt on top of that and replant.

Another way would be to bring in 5 yards of topsoil and raise the grade of the area to push that water else where. Unfortunately it looks like it would be sent directly into your neighbors back yard.

Not sure if they'd be too fond of that.

I've got one other way that I've seen work and that's simply laying sod directly on top of the existing ground. It raises the grade just a bit and soaks up a fair amount of water too.