How big is the area?

How close is it to buildings?

Is it soggy year-round?

What is it used for?

What's around it?

You can turn it into a bog garden and make it incredibly beautiful and not have to mow at all.

Dig a pond.

The idea of the pond that connects to the neighbor's is excellent.

Is it in desirable grasses, like fescue?

Or is it in weeds right now?

There is nothing wrong with letting the grass grow high, unless it's an undesirable grass.

In fact, that's one of the best ways to handle it.  Let that wet spot grow high, and mow around it to the point that it starts to feel spongy.

The one thing about herbicides is that it's very difficult to find any that are not problematic in wetland areas.

You have to be really specific with the target, and it has to be a narrowly targeted application, in most cases, if it's anywhere near a wetland.

Now...maybe you don't care, but I'm telling you what the labels say (generally) and that if you can find a natural approach to it, since you plan to dig it out as a pond later anyway, you'll do well to choose that approach.  Your neighbor may appreciate it too, since in many cases like yours,  water is going to mix with water, either above ground or underground.  

Of course, you may take a nuke it from orbit approach with your belief system, and if so, YMMV.