Some of this functionality is provided OOTB, but with configuration on the part of the System Administrator, not necessarily the user, and some of what you're asking for requires customization.
There are 3 ways in which you can build notification systems in Dynamics CRM 2011.
1.) Workflow
2.) Activity Feeds
3.) Custom Dev/3rd Party Tools
Option #1 will require a bit of configuration. You would need to allow a Workflow to be triggered when an Activity is created where the Regarding field is a Case. This is fairly easy to achieve.
However, getting it to send out the notifications at a particular time of day would require the workflow to execute on a regular schedule. This would require code. We have a Workflow Scheduler that allows you to execute a workflow (such as an email notification) at a regular interval (minute, hour, day, month, etc.). It would be very easy to allow the users to configure their own scheduler, they would simply need to select the workflow (That the system admin would create) and the time interval.
Option #2 uses the native Activity Feeds (which are similar to displaying posts on your Facebook wall), only inside of Dynamics CRM. Future versions of Dynamics CRM will phase out the use of the Native Activity Feeds and use Yammer instead.
Here you have a little more flexibility in how the user choses to receive notifications, yet you still would not be able to schedule them for a particular time of day.
With a little configuration, though, you can automate "posts" to the users wall (and send out emails, too, if you so choose), based on similar criteria. One benefit to Posts as opposed to Emails is that they don't clutter the inbox. A downside is they don't get Push notification (unless you send out emails as well).
Option #3 would of course require some investigation, but a good place to start would be c360's Alert product, though that may not be as flexible as you need it to be.
