What's the best way to back up a Google Apps user's emails and other data before deleting their account?

There are great third party backup tools available in the Google Apps marketplace but they may not be the best solution depending on how active the user account data is.

Make sure you ask yourself and understand the reasons you are backing it up in the first place.

Back-ups can lock your data away and make it hard to access through multi-step processes you may not have time for.  Deleting a user's account may not be the answer either.

If you think you will need to access the data sometime in the near future, it may be best to leave it on google's servers so it can be quickly accessed in a few clicks rather than removing it and storing it in a less accessible and un-searchable manner.

There are a few ways to do this.

One of the easiest ways of quickly archiving an old user's data is by suspending the account. This will keep the old user from using the account and keep the data safe on Google's servers.  This is reccomended as a way to shut down account access quickly.  It's actually a temporary measure if the data in the account is routinely needed. 

Learn how to suspend a user here:http://support.google.com/a/bin/...

For more active information:

Another great method is re-naming the user. 
When someone moves on from a position they are often replaced.  If the replacement person is new, it benefits the new person to have access to the old person's data. Re-naming the user gives them access to all the files and information their predecessor had and creates an alias email address on the account so they will continue to get all the email being sent to the old person.

Learn how to re-name a user here:
http://support.google.com/a/bin/...

Sometimes the information in an account needs to be accessed by a supervisor or existing person rather than a new user.

For information stored in Google Docs you can transfer ownership of all documents from one user to another:
https://support.google.com/a/bin...

For email information, you can check out the email migration options here:
http://support.google.com/a/bin/...

There are also other methods of extracting user data here:
http://support.google.com/a/bin/...