You can do it by maintaining the values at class side

Create new varaible; private List<SelectOption> options;

initialize it in constructor at 1st line; options = new List<SelectOption>()

then from this method getName remove this line List<SelectOption> options = new List<SelectOption>(); and you are done with your solution.

earlier what was happening, when ever you was changing the values of Country, your options List was re-initializing every time.

Please refer this code for your class. I have created a new String variable strGeographix which will store geograpic values and put in your errorMap on every call.

    public class RfpControllerExtension2 {

public string model{get;set;}
public List<String> SelectedDept{get;set;}
public List<Geography__c> results{get;set;}
public List<SelectOption> AllDept{get;set;}
public List<SelectOption> AllName{get;set;}
public List<String> SelectedName{get;set;}
public String discountScheduleID { get; set; }

private String strGeographix = '';

public RfpControllerExtension2 (){
    AllDept= new List<SelectOption>();
    AllDept= getDept();
    AllName= new List<SelectOption>();
    AllName= getName();

}

public void FindAllName()
    {
        system.debug('@@@@@@ Entered');
        AllName= new List<SelectOption>();
        AllName= getName();
    }  

public List<SelectOption> getDept()
    {
        List<SelectOption> options = new List<SelectOption>();
        options.add(new SelectOption('Select','--Select--'));
        Schema.DescribeFieldResult departfield =  Geography__c.Country__c.getDescribe();
        List<Schema.PicklistEntry> picklist = departfield.getPicklistValues();
        for( Schema.PicklistEntry f : picklist){      
        options.add(new SelectOption(f.getLabel(), f.getValue()));
        }    
        return options;

     }

   public List<SelectOption> getName()
    {
        List<SelectOption> options = new List<SelectOption>();
        Set<String> sProj = new Set<String>();
        
        if(discountScheduleID != null )
            {
               for(Geography__c Proj  :[select Id, Canonical_Name__c, Type__c from Geography__c where status__c = 'active' AND Country__c =: discountScheduleID order by Canonical_Name__c] )
               {
                    options.add(new SelectOption(Proj.ID,Proj.Canonical_Name__c));
                    strGeographix += Proj.Canonical_Name__c + ', ';
               }
            }
            errorMap.put('selectedGeographies',Proj.Canonical_Name__c);

        return options;

     }

     public Map<String, String> errorMap {
        get {
            if (errorMap == null || errorMap.size() == 0) {
                errorMap = new Map<String, String> {
                    'selectedGeographies' => '',
                    'flightDates' => '',
                    'agency' => '',
                    'agencyContact' => ''
                };
            }
            return errorMap;
        }
        private set;
    }

}