I recently saw this in Red Dead Redemption 2, which is set in the early 20th century when the device is seen in a bank.
I’ve also seen something similar in the old Davy Crockett movie set in the early 1800s.
I believe I’ve seen it in other westerns, but none I could name off the top of my head.
I’m just very curious and I’ve had no luck googling this topic. It’s kinda hard to google “thingy from westerns help”.
What you're describing is something that was common before the invention of the fountain pen. It's an ink blotter. Before fountain pens one had to dip the "nib" or writing point of a pin into a separate ink well before writing. This would typically leave excess ink on the paper that was slow to dry and easily smeared if handled too quickly. The blotter was used to soak up the ink without smearing the writing. It has a curved, semi-cylindrical shape so that it can be rolled over the writing and not smear it. I have one left to me by my great-grandfather.
It would help if you could provide a screenshot.