How can daily brightness be quantified or measured?
It can be measured in units of energy per surface area, for example, kWh/m2.  The official SI name would be radiant exposure but I haven't seen that phrase used in a climate or weather context.

For example, here is a map by NREL of average radiation per day in the US for July:


(Source: National Renewable Energy Laboratory)

And here is a map for Sweden for the same quantity per year (hence the much larger numbers):


(Source: SMHI)