> This may be a silly question, but how much weight would it take to cause this?

Mathematically, there's not a minimum threshold, but in practice, there's going to be mass distributions that are going to produce such a small predicted deflection they are not really measurable. The math for flexure is laid out in a variety of places, Wickert, 2016 provides a pretty complete view if you can't get your hands on a copy of Turcotte & Schubert. Thus, you can calculate the predicted flexure for any mass, but in practice, that mass may be insufficient to produce a measurable flexure. The other big complication here is that the response also depends on the duration of the load and/or the rate of change of the load through time as the way the lithosphere responds to loads (i.e., purely elastic, viscoelastic, etc.) depends on the rate of change of the load (e.g., Watts et al., 2013).

> Also, does it depend on the makeup of earth that sits below said weight?

Yes. If you look through the math in Wickert, you'll see a few terms that potentially vary with location, specifically the density contrast between the infilling material and the mantle and the flexural rigidity (D). For the former, this means that the density of the load (i.e., is it rock, water ice, liquid water, etc) matters, but also that theoretically the density of the mantle in that location matters. In practice, we often assume a standard density for the mantle (not necessarily always) so we don't often consider this term to vary by location (but in reality, it might). However, flexural rigidity definitely does vary by location. If you go to the appendix, you'll see a definition for D that includes Young's modulus, the Poisson ratio, and the effective elastic thickness (Te). We typically assume Young's modulus and Poisson's ratio are constants for the lithosphere, but Te can vary a lot by location (e.g., Watts, 1992, Burov & Diament, 1995, Burov, 2011), e.g., the oceanic lithosphere generally has a narrow range of Te with most being ~10-20 km whereas continental lithosphere has a pretty wide range of Te with some in similar ranges as oceanic lithosphere but others being significantly thicker. The effective elastic thickness is kind of what the name implies, i.e., it's an approximation of the thickness of a purely elastic sheet that would explain the observed deflection for a given mass distribution. Te is generally not a physical thing (many of the cited papers are trying to find relations between Te and something we can actually measure like crustal thickness, temperature profiles, age, etc) but is something we estimate from observed deflections (though for oceanic lithosphere, it is more explainable as a function of lithosphere age/temperature). In general, for the same surface mass and mass distribution lower Te means more "local compensation", i.e., larger deflections with much shorter wavelengths, whereas larger Te means less deflection distributed over a much longer wavelength. In practice, Te is the main thing that we consider to vary as a function of location (and in turn, flexural ridigity) and this has a pretty important influence on how that area responds to a given load.