Question
How do you eliminate a woodpecker from pecking through a stucco finished home and nesting in the Styrofoam?
This is my neighbor's home , which looks like swiss cheese as he tries various suggested measures (From the internet wildlife control), that only result in the woodpecker invading a new location...It is fall in NJ, while this is occuring.  

Answer
Not sure what you meant by internet wildlife control, perhaps you meant that he was following control information found by searching the net?

If the woodpecker has actually established a nest, versus just making a hole, then your neighbor has problems. He will need to contact state wildlife officials and obtain necessary permits for lethal control.   If it is just holes, then it appears he has followed hazing protocols and he should then consult state officials to obtain permits for lethal control.  A summary of hazing techniques can be found below.  

LAWS concerning woodpeckers
Woodpeckers are a federally protected bird under the North American Migratory Bird Act. Do not use lethal control on woodpeckers without contacting your Federal Wildlife Officer. You will need to institute non-lethal control strategies before you will receive permission to implement lethal control.  (See below for information on non-lethal woodpecker control). Woodpeckers are a Federally Protected Bird To visit the Law Click Laws
Stopping Woodpeckers from damaging your home
Why Woodpeckers peck your home
Woodpeckers peck at houses for basically three possible reasons,
?      searching for bugs (rare) but consider it if occurring during the winter (Dr. Doug Mampe Pest Control Mag. Feb/02. p.14.),
?      building a home (rare) but if it happens it will also be in the spring or
?      marking territory (common a phenomenon called drumming (During month of March/April/May).
Surfaces Known to have been pecked by Woodpeckers (we welcome additions just e-mail them to us at [email protected] Aluminum flashing: particularly if there is a void behind it.
?  Cedar siding
?  Chimney Caps
?  Fir siding (white fir)
?  Pine Siding
?  Redwood siding
?  Roof vents
?  Space Shuttle
?  Stucco
?  Synthetic Stucco Polymer Coatings (eg; Dryvitt and Stucco-Flex)
?  tile roof
When Woodpeckers peck at shiny objects: "I have a woodpecker problem that I haven't seen addressed on your site. We have one that appears only in the morning, approx. 6:20, and he's pecking on the metal flange on the top of the fireplace and another metal pipe on the roof. The literature that I've read says that they don't like shiny objects, but that's exactly where he's pecking!"
ANSWER: He is probably drumming (he likes the sound) or he sees his reflection. Just because it is metal doesn't mean it is shiny enough. The shiny I am talking about is mirror shiny. If he still pecks at it, then it is possibly due to his seeing a reflection of himself and thinking it is another bird. The least expensive option is probably the mylar tape. I would try that first.
Don't Give Up Hope:
In one study the birds stopped drumming 50 percent of the time within two weeks or so whether the homeowners did anything or not.
The best information is found at http://icwdm.org/handbook/birds/Woodpeckers.asp Below will be a synopsis of information found there.
Non-lethal strategies to control Woodpecker damage    
Unfortunately, there is no easy guaranteed solution. So with all that being said, I would suggest the following strategies:
First. Cover all holes as soon as possible. It is recommended that you place aluminum flashing over the areas where the woodpecker is pecking. The flashing will stop the pecking at that spot because 1. its metal and 2. it changes the sound and 3. woodpeckers don't like shiny objects. Just make sure that the woodpecker is not living in your home. Disturbing a woodpecker den or killing one is a federal offense.
Option 1. Harass and scare the woodpecker causing damage using one or more of the following techniques.
Remember no harassment technique works all the time or in every situation.
?  Mylar tape. You can also try running some mylar tape (1 inch wide strips) around the area where he is pecking. Woodpeckers don't like shiny objects. If you don't have mylar, use tin foil or small mirrors.
?  Distress Tapes. There are machines that digitally recreate woodpecker distress calls. These are NOT Ultrasound. When you turn on the device, it spooks the woodpecker.
?  Scary Eye balloons. These balloons mimic the look of an owl which spooks the woodpeckers. Click Balloons
?  Garden Hose: One animal damage controller says "Pre position a garden hose with a sprinkler set up on an angle to reach where the bird is drumming. Turn on the hose and watch the woodpeckers leave. It only takes a few squirts. They leave quickly and don't like hanging on to a wet siding etc.. " There is an automatic sprinkler on the market called the Scarecrow
?  Attack Spider: This is a relatively new (2003). It activates using a sound detector to scare woodpeckers through sight and motion.
DON'T USE
?  Owl Effigies: unless you are willing to move it around on a daily basis. Understand that at best the effigy will work only in the short term if at all.
?  Ultrasound: if you can抰 hear it, neither can they.
Option 2. Employ exclusion techniques. If woodpeckers are damaging your siding under an eave, hang some netting from the eave line down to the ground. If the net is extended away from the house wall, the woodpecker can't get close enough to damage the wood. Some homes actually leave the hooks up year round and then hang the netting as needed.  This is a great solution as it is permanent! However, it shouldn抰 be used if the woodpeckers have decided to reside in the hole they created.
I would also encourage you not to dally. You want to act quickly before the woodpecker decides your home is a nice place to live.
Option 3. Only to be used after one of the previous options has been tried and failed.
Keep records. Lethal Woodpecker Damage Control Strategies  (Federal permit needed and sometimes a state permit)
Visit http://icwdm.org/VendorsService/default.asp click the state you live in to find the web site of the state agency in your state. If you need a professional, you may find one listed there too.
?  Shooting. Make sure you follow all relevant Federal, State and local gun use laws.
?  Trapping. Probably the most effective and safe.
Disclaimer: As with any activity, remember that animal damage control comes with its own risks and problems which can include but are not limited to legalities, health threats, and personal liabilities. Be sure to follow all state laws governing wildlife and make sure you have a thorough understanding on how to resolve the animal damage complaint. My advice is only as good as your understanding of me and my understanding of your situation. If you have any questions be sure to write back.