My initial guess: USB for the HDD/SSDs, and the native µSD slot for copying.
Connecting a hard-drive / SSD is not directly possible via SATA, since the Omega2 doesn't have any SATA (m.2) ports.
You can buy a USB harddrive docking station, so attaching SSDs and HDDs will work out-of-the-box and appear as mountable devices. The only programming you'd have to do is a shell script or python script which looks for µSD cards to be copied and hard drive targets. A simple
dd command might be enough.
Since the Omega2 only has one USB 2.0 port (not USB3.0), speed might limited when writing to the harddrives, depending on what is slower (USB or µSD card accss)
If you buy an Omega2+, you have one native µSD card slot. Just put your program /shell script on the main embedded flash and insert the µSD card in the native slot you want to copy from.
For extra gimmicks you might also add LEDs controlled via GPIOs to start blinking/flashing whenever a µSD card is inserted or a copy process starts.