Tom, 

To get the value back into the controller after the lookup field is filled, you need to click a command button. I modified your controller code below. This should work for you.

public class CFCT_ContactOwnerChange_Controller {

    private String currentContactOwnerId;
    private String ownerId;
    private String ownerName;
    private String newOwner;
    private Contact myContact;
	
	//when the controller is called get the contact based on the id param
	public CFCT_ContactOwnerChange_Controller() {
            myContact = [select id, name, OwnerId from Contact where id =
                       :ApexPages.currentPage().getParameters().get('id')];
      }
	
    /* Getter which dynamically generates a unique list  
       of contact owners from the Contact object */        
    public List<SelectOption> getcurrentContactOwners() {      
        List<SelectOption> optionList = new List<SelectOption>(); 
        /* Add a null option to force the user to make a selection. */
        optionList.add(new SelectOption('','--None--'));  
        /* Loop through the Contact records creating a selectOption  
        for each result with the owner Id as the value and the owner name 
        as the label displayed in the selectList */  
        for (AggregateResult c : [ SELECT ownerid, owner.name ownername 
                                   FROM contact 
                                   GROUP BY ownerid, owner.name Limit 500 ]) {
            ownerId = (String) c.get('ownerid');
            ownerName = (String) c.get('ownername');             
            optionList.add(new SelectOption(ownerId, ownerName));  
        }  
        return optionList;
    }   
         
    public void setCurrentContactOwner(String cco) { 
        this.currentContactOwnerId = cco; 
    }    
    
    public String getCurrentContactOwner() {
        return this.myContact.Id;
    }

    public PageReference changeOwner() {
        //PageReference pageRef = ApexPages.currentPage();        
        //************************************************
        // ADD CODE TO UPDATE THE CURRENT OWNER TO THE NEW
        // OWNER FOR ALL CONTACTS
        //************************************************        
         
		PageReference pageRef;
        
		if (this.myContact.Id == null) {
		  //put error on page and send them back to choose a new owner.
          ApexPages.Message myMsg = new ApexPages.Message(ApexPages.Severity.FATAL,'Please choose a Current Owner');  
          ApexPages.addMessage(myMsg);
		  pageRef = null;
        }
		else{
		  //Do Something with new Owner Id
		  //send them back to the contact they came from
		  pageRef = new PageReference('/' + myContact.Id);      
		}
        return pageRef;
    }
   
   public Contact getNewContactOwner() {  
     return myContact;                                          
   }

}

I added a constructor to get the Contact sObject the user came from and changed the changeOwner and getNewContactOwner methods.

hope this helps