(Full image) Beechcraft 1900C.
I cannot find the name of the missing instrument even in the manuals.
Originally there was no instrument at this place, except for UE series with an electrically driven artificial horizon, where there was a fast erect pushbutton/indicator for the copilot gyro.
According to Wikipedia, UD series is for the military model and UE series is for 1900D, so I assume UC series is for 1900C model. Instrument panel for series UA/UB/UC:
Instrument panel from training manual, models UA/UB/UC (source)
Anything visible here has been added after the aircraft left the plant, when the panel was retrofitted. When browsing pictures of this cockpit online, we can see various additions:
Looking at details of your image, we can see there was likely a sticker and a button associated with the instrument. This could fit with a battery temperature indicator and its test button:
This configuration was actually found on F-GPYY. Comparison:
Full cockpit of F-GPYY when it was an aircraft of the now defunct French Air Littoral operator:
Photo by Thierry Izzi, source
It's a missing instrument because it's not part of the standard avionics of the 1900C model but part of the Beechcraft's 1900D aircraft avionics.
If you take a look at pages 0 and 1 of 31-10-00-01 (chapter 31, section 10, unit 00, figure 01) of the Beechcraft 1900D aircraft's IPC (Illustrated Parts Catalog),¹ you'll notice that the hole featured on the picture that you provided is precisely the item
100 (Part Number
101-380031-3), which is the Copilot's Turn and Slip Electric Indicator (technical nomenclature
INDICATOR-COPILOT TURN AND SLIP, ELECTRIC).
Looks like this 1900C aircraft is using a 1900D aircraft panel plate/harness but it doesn't have all the 1900D's indicators, hence the missing item
¹ Pages 6 and 7 of the PDF file.
I flew the 1900D. We had 20 of them in the fleet but there wasn’t a gauge or hole there.