Index: wp-admin/includes/dashboard.php
===================================================================
--- wp-admin/includes/dashboard.php	(revision 9120)
+++ wp-admin/includes/dashboard.php	(working copy)
@@ -109,7 +109,7 @@
 			return;
 		}
 		list($url) = explode( '#', add_query_arg( 'edit', $widget_id ), 2 );
-		$widget_name .= ' <a href="' . clean_url( "$url#$widget_id" ) . '" class="open-box">' . __( 'Edit' ) . '</a>';
+		$widget_name .= ' <a href="' . clean_url( "$url#$widget_id" ) . '" class="edit-box open-box">' . __( 'Edit' ) . '</a>';
 	}
 	add_meta_box( $widget_id, $widget_name , $callback, 'dashboard', 'normal', 'core' );
 }
Index: wp-admin/css/dashboard.css
===================================================================
--- wp-admin/css/dashboard.css	(revision 9120)
+++ wp-admin/css/dashboard.css	(working copy)
@@ -1,3 +1,11 @@
+.edit-box {
+	display: none;
+}
+
+h3:hover .edit-box {
+	display: inline;
+}
+
 form .input-text-wrap {
 	border-style: solid;
 	border-width: 1px;