I started by having a look in the list of devices that are available for passthough. The Lynx Point AHCI Controller was not on the list of devices since SATA controllers are unsupported for passthrough. Let's fix that!
Log in to your host using SSH or use the DCUI locally.
Find out the PCI ID using the following command:
esxcli storage core adapter list
Your onboard SATA controller is usually listed as vmhba0. The ID we're looking for is listed in the Description. So in this case, it's 0000:00:1f.2.
Enter the next command to find the PID. Substitute the PCI ID with your own ID if needed.
lspci -n | grep 0000:00:1f.2
In this case, the PID we're looking for is 8c02. Add this PID at the bottom of /etc/passthrou.map like so.
Save your changes and reboot the ESXi host.
After the reboot, you'll be able to select the Lync Point AHCI controller.
So after this, follow the normal procedure for adding a passthrough device to a VM.
Mark the device for passthrough and reboot the host.
When the host is up again, edit the VM settings to add the PCI device.
In my case, the VM I added the controller to is a Windows 7 virtual machine. To show you the controller and the connected disks, I added a screenshot of the device manager. I have two Samsung 850 Pro SSDs connected to the onboard SATA controller.
Many thanks to Hilko for showing me the commands in the "Energy efficient ESXi server" thread on got.tweakers.net. Most of this blogpost is a shameless copy of his post.