Why use xenon in the search for dark matter? The XENON dark matter research project is an interesting long-running project that strives to detect dark matter. I understand that the basic principle of the project design is that WIMPs are expected to "flash" and "ripple" when they interact with an atom's nucleus, as detailed in this Physics post. 
My question is what makes liquid xenon the best medium for this? I understand that xenon is extremely low reactivity and has a really long half-life, but it seems to me you should want a highly reactive medium if you are hoping to find weakly interactive particles. Low reactivity seems like the wrong tool. What am I missing here? 
 A: The liquid xenon is serving a dual purpose.
First, the xenon nuclei serve as a target for the WIMPs to interact with.  There are a number of hand-wavy reasons why heavy nuclei would be more likely to interact with most kinds of WIMPs than low-mass nuclei:  more mass density, more charge on the nucleus, more electron density in case the WIMP interaction is with the electrons rather than the nucleus, and probably some other reasons.  The details, and the theoretical models which justify them, occupy a chapter in most PhD dissertations on WIMP searches.
Second, the liquid xenon acts as a scintillator, transforming deposited energy into light which can be carried to a photodetector.  This is true regardless of the source of the ionizing radiation: the xenon also scintillates in response to cosmic ray muons and intrinsic radioactivity.
Noble liquids are nice to work with experimentally because they are self-purifying:  any chemical contaminant which is not xenon will freeze out of the liquid.  All of the noble liquids have similar chemistries and behave as scintillators.  (There are some neutron experiments which use liquid helium as the same kind of target-plus-detector combination.)  But if increasing the nuclear mass increases the cross-section for interacting with WIMPs, you want your target to be made of the heaviest available nucleus.  And the heaviest non-radioactive noble gas is xenon.
A: Short answer: because it works very well.
There's not just one reason of why xenon is a good choice but rather a series of them. These conspire to make liquid xenon time projection chambers the leading technology to directly search for rare signals, used not only by the XENON collaboration, but also by LUX/LZ, Panda-X, and EXO/nEXO.
Some practical advantages in no particular order:

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*Like other noble gases (well, noble liquids... noble elements... you know. Noble Gases with a capital "G"), xenon can be purified well from other contaminants. This is important since these contaminants can be radioactive, or can negatively impact the operation of the experiment due to their chemistry.

*If you look at some older presentations you will see these note that xenon contains no radioactive isotope. This is in contrast to e.g. argon or krypton, where naturally occurring radioactive isotopes create a huge amount of background events. This statement is now technically no longer correct, since EXO discovered the "two-neutrino double-beta decay" of xenon-136 and XENON1T discovered the "two-neutrino electron capture" of xenon-124 but both these decays have crazy long half-lives, not comparable to usually naturally occurring radioactivity.

*Xenon is very dense (or, more precisely, has a high nuclear charge Z). That implies that it is a very good shield against radioactivity, in particular gamma rays. Any radioactivity coming from detector construction materials thus tends to get stuck in the outer few centimeters of the detector target. The innermost volume is then very radio-clean and gives even better sensitivity to rare signals.

*Xenon is an extremely bright scintillator. This gives a very low energy threshold, which increases the sensitivity of the experiment.

*Xenon scintillation light can be read out with dedicated photomultipliers, without a need for wavelength-shifters

*Xenon is liquid at about -100 degrees Celsius. Thus, cryogenics are easy, no need e.g. for complicated dilution refrigerators.

These are all instrumental advantages. Further,

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*Xenon is relatively cheap, at a couple or at most a few 1000$/kg of target material. This is much cheaper than competing technologies that use e.g. high-purity germanium crystals or isotopically purified argon. In turn, one can afford even multiple and large experiments.

There are also real dark matter-related reasons though:

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*For standard WIMPs of tens or hundreds of GeV in mass (tens or hundreds of proton masses), historically one would search for spin-independent scattering, i.e. coupling of the dark matter to the nucleons in the nuclei. In this case one expects the wavelength of the dark matter to be so long that it can not resolve the nuclear structure of the nucleus in the scattering. Thus, the interaction is said to be "coherent", in which case the scattering cross section (a measure of the interaction probability) scales like the mass number A of the target nucleus squared. With xenon at the bottom right of the table of elements, that gives a boost in sensitivity when compared to lighter target materials.

*Half of naturally xenon is in "odd" isotopes, i.e. with unpaired neutron spin. That gives xenon even sensitivity to spin-dependent couplings.

*Xenon being heavy means that in "inelastic" dark matter scenarios, higher mass splittings are kinematically accessible. Though these models are certainly not mainstream, it's a nice-to-have feature.

Regarding "reactivity": noble gases are chemically inert (mostly). But these experiments do not look for dark matter chemically reacting with the target. Instead, they search for billiard-ball-like scattering events, of the sort that radioactivity would induce. One commenter writes "low reactivity [...] should be read as low chemical reactivity" -- no, it should be read as "low radioactivity".
