Is it possible to install OS X Mavericks from one Mac to another?
You can't "install" it, but you can clone an image of the OS on your laptop using Disk Utility and save it to an external hard drive, then restore that image (also with Disk Utility) onto your Mac mini.

Even better, if you can use Target Disk Mode (requires both Macs have either Thunderbolt or FireWire), you can use the same restore feature in Disk Utility to clone directly between the drives without requiring the external media/image file.

This will of course wipe your Mac mini drive in the process and won't give you a clean install.  If you're OK with that though, then it might be an option if you really can't afford the download.