In an ADC what are the units of resolution called?
Speaking as an applications engineer at a company that makes ADC chips... the unit is called LSB (least significant bit), so in your example the ADC output code would be 4000 LSB. (A commenter notes that this could also be called Counts; this would also make sense for dual-slope ADC.) 

This is normally only interesting when reading the ADC's datasheet, where there's usually a specified level of accuracy for INL (Integral Non-Linearity) / DNL (Differential Non-Linearity) / offset error / gain error. See for example the datasheet for MAX11300,  the ADC Electrical Specifications. (Disclosure: I am a Maxim applications engineer, and this is one of the products I support.) Maxim gives the INL/DNL units as LSB.

Note this isn't a physical unit, it's just a name for the raw count value returned by the ADC. There doesn't seem to be a real consensus as to what to call this raw ADC code unit. Typically an application would use some linear mapping (and possibly calibration values) to convert the ADC LSB code into a meaningful physical unit. For example, 4000 LSB * 5.00 Volts / 4096 LSB = 4.88 Volts.

When I write software to do this kind of conversion, I usually name the raw ADC value something like ADC_Code and the voltage as ADC_Voltage. I would be concerned that in the context of software, "count" could be mis-interpreted as an iteration count (which is valid inside a DVM or other dual-slope ADC that actually counts how long it takes to charge/discharge a capacitor).