6 of 14 added 84 characters in body

A little but time consuming experiment first. You'll not need to do it, as it is only for finding a model.

Let's see how the Min value of the plot intensity varies with Opacity:

theData = RandomReal[NormalDistribution[], {2000, 2}];
f[x_] := f[x] =
Min[Norm /@ Flatten[ImageData@Rasterize[
ListPlot[theData, AspectRatio -> Automatic, ImageSize -> 200,
PlotStyle -> {Black, Opacity[x]}, Axes -> False]], 1]];
Plot[f[x] , {x, 0, .4}, PlotRange -> Full]


So, it is an exponential.

Let's fit it:

data = Table[{i, f[i]}, {i, 0, 1, .1}]

model = a Exp[b x];
fit = FindFit[data, model, {a, b}, x];
modelf = Function[{t}, Evaluate[model /. fit]]

Show[ListPlot@data, Plot[modelf[x], {x, 0, 1}]]


Now you are ready to set the min value of the brightness of the plot to whatever you want:

(The Sqrt@3 is a normalization factor for the intensity of the {1,1,1} RGB pixel.)

Let's use it:

opac = x /. Solve[# == a E^(b x)/Sqrt@3, x] /. fit & /@ {1/2, 1/4, 1/20, 1/200}

ListPlot[theData, AspectRatio -> Automatic, ImageSize -> 200,
PlotStyle -> {Black, Opacity[#[[1]]]}, Axes -> False] & /@ opac