So in my onUpdate method in my AppWidgetProvider class, I ended up executing a non-trivial amount of code so that I can completely recreate a new RemoteViews object. The reality
You can update the remote views and then call
ComponentName componentName= new ComponentName(context, YourClass.class);
AppWidgetManager.getInstance(context).updateAppWidget(componentName, remoteViews);
on, which AFAIK should update the widget
First, RemoteView is not a View. It's a set of instructions that build a View hierarchy. It is used to recreate a View in another process (App Widgets do not execute in your app's process). As such it's serializable and mutable.
So, when you initialize a RemoteView just store a reference to it somewhere, e.g. in a field of your custom AppWidgetProvider. Next time you need it, get it from field and the change something on it. For changing the string in a TextView use setString(..).
remoteView.setString(textViewId, "setText", "some text for TextView")
This is the 2013 update if you are using current API's. In your WidgetProvider class' method that will perform an update:
AppWidgetManager appWidgetManager = AppWidgetManager.getInstance(context);
rv = new RemoteViews(context.getPackageName(), R.layout.widgetlayout);
rv.setTextViewText(R.id.ofTextViewInWidgetLayoutXML, "Hello World");
appWidgetManager.partiallyUpdateAppWidget(appWidgetIds[i], rv);
Note that it is no longer remoteView.setString but remoteView.setTextViewText