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A little experiment first. Let's see how the Min value of the plot changes with Opacity:

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:

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:

opacity = Log[ MinDesiredIntensity /Sqrt@3 / a] / b

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