Is there a product that is a mat that can keep soil in place but allow water to seep through?
There are several such products, but before advising, I would want to know what will be under your soil? Airspace? More soil? What is the size of your project in foot dimensions? Is there a slope?
If you have a seeping slope of some size that could slip onto adjacent public or private property you may need a more substantial solution perhaps involving a landscape architect or other certified professional who can come up with something that meets approval of any interested governing agencies.
I have standing water around my lawn, and the lawn is rather wet. I like to dig a small trench around part of the lawn (Maybe about 6 inches) and allow the water to drain down a slope. I could use rocks to place around the lawn and into this small trench, but wondering what the cost would be. I would need about 30 feet by 2 feet of material.
if u have access to a pickup... $50 or so worth of drain rock could do it. depends on depth.
bigger is better with drainage.
Do not direct excess water onto neighboring property. You will first have to identify someplace for it all to go, on your own land.
I might look to address this issue somewhat differently....have you considered a rain garden? A rain garden is intended to direct excess rainfall to an area on the property where it can naturally dissipate or be absorbed. Rain gardens can be constructed in the low spots where rain puddles naturally or you can direct this water via rocked channel (ala 'dry stream bed') to a more suitable rain garden area.
WSU (our extension service sponsor) has developed some excellent resources for homeowners to address these drainage and run-off issues with the minimal amount of environmental impact. There are also a fair number of designers and landscaping companies around specializing in rain gardens if you feel a DYI project too daunting.
I would certainly suggesting investigating this process a bit before you decide how to proceed. In some areas (Kitsap co. for example), county government is offering cash incentives to install and plant rain gardens.
Here is a link that might be useful: WSU rain garden info
A picture or two would be worth a thousand words.
As it is, there's not enough information for a viable solution.