Because of the advantage of USB 3.0 Raspberry Pi 4 Modell B is recommended. It shouldn't make a big difference whether you use the 2GB, 4GB or 8GB variant here.
Some other Raspberry Pi's were successfully tested. So far, only the Raspberry Pi 1 has proven to be really unusably slow, at least if you want to use a display.
Unfortunately, there are no reports about other single-board computers. Experiences are welcome to be reported in the discussions.

Computer	Experience
Raspberry Pi 4 Model B	No problems with all variants
Raspberry Pi 3 Model B	About half the copying speed compared to the Raspberry Pi 4 thanks to USB 2.0
Raspberry Pi zero 2 W with USB Hubs HAT	About half the copying speed compared to the Raspberry Pi 4 thanks to USB 2.0
Raspberry Pi zero with USB Hubs HAT	Slow responding web GUI, even lower copying speed compared to the Raspberry Pi zero 2 W

Camera manufacturers sometimes format memory cards in proprietary formats, which can make handling difficult. On the other hand, there are also mass storage devices that cannot be operated directly on the Raspberry Pi in a stable manner, for example because of their high power consumption. Basically, however, it can be assumed that almost all camera models and almost all storage devices can be operated with the Little-Backup-Box, albeit with a little effort. The compatibility list is a start and will grow with every feedback.

Manufacturer	Model	via cardreader	Camera/direct	Comment
DJI	DJI Mini 2	yes	?
GoPro	Hero5	yes	?
GoPro	Hero8	yes	?
Nikon	Z6	yes	yes	The energy requirement can be too high for the Raspberry pi and especially for weak power supplies, especially in combination with energy-intensive storage media (backup target). A generously dimensioned power supply unit (> 5V / 3A) or, even better, a powered hub enable stable use.
Olympus	E‑M1 Mark II	yes	?
Olympus	E‑M1 Mark III	yes	?
Olympus	E-M5 Mark III	yes	?
Olympus	LS-P4 Hi-Res Audiorekorder	yes	?
Zoom	Zoom H1 Portable Recorder	yes	?
Zoom	Zoom H1n Portable Recorder	yes	?
Zoom	Zoom H4 Portable Recorder	yes	?

Manufacturer	Model	via USB/cardreader	Comment
?	UHS II SD Card	yes
Lexar	1667x 128GB SDXC UHS-II card	yes	tested with Ugreen SD / microSD Card Reader and powered hub
Samsung	Portable SSD T5 2TB	yes	powered hub needed
Sandisk	Extreme 128 microSD	yes	tested with Ugreen SD / microSD Card Reader and powered hub
Sandisk	Sandisk Extreme Portable SSD 1 TB	yes
Transcend	700S R285/W180 SDXC 64GB, UHS-II U3, Class 10	yes	tested with Ugreen SD / microSD Card Reader and powered hub

Some devices have an energy requirement that exceeds the capabilities of the Raspberry Pi 4. Problems occurred mainly in connection with XQD and CFExpress cards. Powerd USB hubs can enable reliable operation.

ORICO 4-Port USB3.0 HUB is powered by USB-c and works very well with XQD-Cardreaders.

UGREEN 4-Port USB 3.0 HUB is not tested yet

Stefankante described an interesting hack to stabilize power supply: Speed boost with USB power modification.