You could try using Isolated Storage:
private void SaveGraphics_Click(object sender, RoutedEventArgs e)
        {
            try
            {
                //Save the graphics layer
                GraphicsLayer graphicsLayer = MyMap.Layers["StatesGraphicsLayer"] as GraphicsLayer;
                GraphicCollection pGC = graphicsLayer.Graphics;
                //Isolated Storage -- Write Graphics
                IsolatedStorageFile isf = IsolatedStorageFile.GetUserStoreForApplication();
                
                using (Stream pStream = isf.OpenFile(_ISFileName, FileMode.Create, FileAccess.Write))
                {
                    StreamWriter pStreamWriter = new StreamWriter(pStream);
                    foreach (Graphic g in pGC)
                    {
                        if (g.Geometry is ESRI.ArcGIS.Client.Geometry.MapPoint)
                        {
                            MapPoint pPoint = (MapPoint)g.Geometry;
                            pStreamWriter.WriteLine("Point, " + pPoint.X.ToString() + ", " + pPoint.Y.ToString());
                        }
                        if (g.Geometry is ESRI.ArcGIS.Client.Geometry.Polyline)
                        {
                            ESRI.ArcGIS.Client.Geometry.Polyline pPolyline = (ESRI.ArcGIS.Client.Geometry.Polyline)g.Geometry;
                            ESRI.ArcGIS.Client.Geometry.PointCollection[] pPC = pPolyline.Paths.ToArray();
                            pStreamWriter.Write("Polyline, ");
                            foreach (MapPoint p in pPC[0])
                            {
                                pStreamWriter.Write(p.X.ToString() + ", " + p.Y.ToString() + ",");
                            }
                            pStreamWriter.WriteLine();
                        }
                        if (g.Geometry is ESRI.ArcGIS.Client.Geometry.Polygon)
                        {
                            
                            ESRI.ArcGIS.Client.Geometry.Polygon pPolygon = (ESRI.ArcGIS.Client.Geometry.Polygon)g.Geometry;
                            ESRI.ArcGIS.Client.Geometry.PointCollection[] pPC = pPolygon.Rings.ToArray();
                            pStreamWriter.Write("Polygon, ");
                            foreach (MapPoint p in pPC[0])
                            {
                                pStreamWriter.Write(p.X.ToString() + ", " + p.Y.ToString() + ",");
                            }
                            pStreamWriter.WriteLine();
                        }
                    }
                    pStreamWriter.Close();
                    pStream.Close();
                }
            }
            catch
            {
                MessageBox.Show("Error", "Error", MessageBoxButton.OK);
            }
        }