Most thin clients use a DOM (Disk On Module) to carry the system firmware. Generally these plug directly into a standard 44-pin IDE interface. An example of a fitted DOM is shown on the right. In general the IDE interface can generally be found close the edge of most thin client motherboards.
One important thing to note - which can be seen in the photographs below - is that: Looking at the IDE interface with the board edge behind it, pin 1 is always at the right-hand end of the connector. In the example below this is indicated by the diagonal corner on the white printed rectangle surrounding the IDE connector. (You can also spot the printed '2' to the right aligning with pin 2 of the connector).
If you want to fit a larger DOM you can often find they cost more than you paid for the thin client in the first place! I tend to use Compact Flash cards as (a) The digital cameras I've used over the years have used them so I now have plenty of spare cards of varying capacities lying around; and (b) they're relatively cheap.
Checking on eBay you'll find that there are plenty of Compact Flash adaptors available, both for 40-pin and 44-pin IDE interfaces. In the former case they'll be an additional socket for providing power to the card. This is not necessary with the 44-pin connector as power is supplied through the standard interface. Three typical 44-pin examples are shown below (top and bottom views):
Looking at each in turn:
Whatever Compact Flash adaptor you find and try and use, do check that you connecting pin 1 on the adaptor to pin 1 on the IDE interface. Most thin clients just use two rows of pins on the motherboard without any surrounding socket or keyway. In those circumstances it is easy to get it wrong.
NB: I have no idea why the manufacturers in the Far East do not make a Compact Flash adaptor that mirrors the standard DOM in its physical orientation. Maybe they will someday, but for now, every one I've seen advertised on eBay has been back-to-front as far as our needs are concerned.