Deleting is secure when the files that were deleted get immediately written over with 0s and 1s, making any recovery attempt for those files harder.
If those files you want deleted over the network would be e.g. where your Time Machine backups are, just install the OS again on the system you are giving away, and don't grant it access to the system.
As you are giving the Mac Mini to your son, I would recommend doing a clean Mac OS X install on that Mac before you create him a user. While you could just delete all your user specific files, and your old user, and create him a new admin user, it is still a better practice to wipe the system clean and give him a whole new start with the Mac.
All you need is the install discs (that came with the Mac, or 10.6 discs if you got a retail version after), and less than an hour of time - installing it is easy and I wouldn't recommend skipping it, provided you have the discs available.