Added functionality for showing new popup window as a popup box, and also for closing this popup box automatically whenever it loses focus (goes into background). Thereby making it more similar to standard Salesforce popup box which appears while assigning a new account to a contact.
// The controller for main VF page vfAssignNewAccountToContact
public with sharing class AssignNewAccountToContact
{
    public Contact c {get;set;}
    public String selectedAccountName   {get;set;}
    public String selectedAccountId {get;set;}
 
    public AssignNewAccountToContact()
    {
        selectedAccountName = '';
        selectedAccountId = '';
        c = [select id, name from Contact where id = :ApexPages.currentPage().getParameters().get('id')];
        System.debug('////////////c is ' + c);

    }
    
    public pageReference changeTheAccount()
    {
        System.debug('//////////selectedAccountId is ' + selectedAccountId);
        System.debug('//////////selectedAccountName is ' + selectedAccountName);
       
        if(selectedAccountId != '')
        {
            System.debug('//////////selectedAccountId is not null, so proceeding ');
            c.accountId = selectedAccountId;
            update c;
            ApexPages.addMessage(new ApexPages.message(ApexPages.severity.info, 'New account assigned successfully'));
            
            /*
              Make variables null, so that if something is typed now in the textbox,
              it gives error message if "Assign new account" is clicked again.
            */
            selectedAccountId = '';
            selectedAccountName = '';
        }
        else
        {
            System.debug('//////////selectedAccountId is null');
            ApexPages.addMessage(new ApexPages.message(ApexPages.severity.error, 'Kindly select an account by clicking on Lookup'));
        }
        
        return null;
    }
}

<!--
  VF page name :- vfAssignNewAccountToContact
  Main VF page for assigning new account to a contact. Go to an existing contact record and add
  /apex/vfAssignNewAccountToContact?id=<ContactRecordID> to the URL to view this VF page.
-->
<apex:page controller="AssignNewAccountToContact">
  <apex:form >
    <apex:PageMessages id="messages"/>
    Click on Lookup to select a new account for this contact and then click on "Assign new account"
    <apex:pageBlock title="Contact Details">
      Contact name: {!c.name}
      <br/>
      <apex:pageBlockSection >
        <apex:inputText value="{!selectedAccountName}" id="accountName" label="Account"/>
        <a onclick="popUp();" style="cursor: pointer;">Lookup</a>
        <apex:inputHidden value="{!selectedAccountId}" id="accountId"/>
          
        <script>
            var objInputName = '{!$Component.accountName}';
            var objInputId = '{!$Component.accountId}';
                    
            function popUp() 
            {
              var objAccountInput = document.getElementById(objInputName);
              //alert('parameter is ' + document.getElementById(objInputName).value);
              
              var urlAdd = 'vf_AccountLookup_Controller';
              if(objAccountInput.value.length > 0)
              {
                urlAdd += '?srch=' + objAccountInput.value;
                //alert('urlAdd is ' + urlAdd);
              }
              var newWindow = window.open(urlAdd,'Popup','height=500,width=700,left=150,top=50,resizable=no,scrollbars=yes,toolbar=no,status=no');
            }
        
            function setAccount(recordId,recordName)
            {
              //alert('recordId is ' + recordId);
              //alert('recordName is ' + recordName);
              document.getElementById(objInputName).value = recordName;
              document.getElementById(objInputId).value = recordId;
            }
        </script>
        <br/>
        <apex:commandButton value="Assign new account" action="{!changeTheAccount}"/>   
      </apex:pageBlockSection>
    </apex:pageBlock>
  </apex:form>
</apex:page>

// The controller for the lookup popup box for selecting new account for a contact.
public class AccountLookup_Controller
{
    public String searchText {get;set;}
    public List<Account> searchResults {get;set;}
    
    public AccountLookup_Controller()
    {
        // get search parameter from url if present
        if(ApexPages.currentPage().getParameters().containsKey('srch'))
        {
            searchText = ApexPages.currentPage().getParameters().get('srch');
        }
        else
        {
            searchText = '';      
        }
        searchResults = [Select Id, Name, AccountNumber, Site From Account Where Name Like : searchText + '%'];
    }
    
    public void searchRecords()
    {
        searchResults = [Select Id, Name, AccountNumber, Site From Account Where Name Like : searchText + '%'];
    }  
}

<!--
  VF page name :- vf_AccountLookup_Controller
  The VF page for the lookup popup box which appears for selecting new account for a contact.
-->
<apex:page controller="AccountLookup_Controller" showHeader="false" sidebar="false">
  <apex:form>
    <script>
      window.onblur = function() {
          // close this popup window if it loses focus, that is, goes into the background
          self.close();
      }
      document.onblur = window.onblur;
      document.focus = window.focus;
    </script>
    <apex:pageBlock>
      <apex:sectionHeader title="Select the account to be associated with the contact"/>
      <apex:inputText value="{!searchText}" onkeyup="searchImmediately()"/> &nbsp;
      <apex:commandButton value="Go" action="{!searchRecords}"  
          rerender="results"/>
      
      <apex:actionFunction name="searchImmediately" action="{!searchRecords}" reRender="results"/>
      
      <apex:pageBlocksection title="Account Search Results" id="results">
        <apex:pageBlockTable value="{!searchResults}" var="a"
            rendered="{!searchResults.size>0}">
          
          <apex:column headerValue="Name">
            <a style="cursor:pointer;"  
                    onclick="window.opener.setAccount('{!a.Id}', '{!a.Name}');
                self.close();">
                {!a.name}
            </a>
          </apex:column>
          
          <apex:column headerValue="Account Number" value="{!a.AccountNumber}"/>
          <apex:column headerValue="Account Site" value="{!a.Site}"/>      
        </apex:pageBlockTable> 
            
        <apex:outputText value="No results found"   
            rendered="{!searchResults.size=0}"/>
      
      </apex:pageBlocksection>
    </apex:pageBlock>
  </apex:form> 
</apex:page>