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In other words, What physical law can X manipulate in order to control the weak force?
And as for the abilities, I...don't have the greatest grasp of what someone can do with the weak force as their lay thing. Out side of causing fission and fusion reactions/explosions and irradiating everything. So some pointers on what else can be done with the weak force.
An extension of two other posts:What would powers based on electromagnetism look like? and What would powers based of Gravity look like?
The most interesting possibility here is nuclear alchemy. If the practitioner can have very fine grained control over the weak force on bulk materials, the possibility exists that they could trigger specific modes of radioactive decay in specific elements, causing them to become other, possibly more useful, (and perhaps even stable) elements. "Impossible" materials which would be too short lived in real life could be made stable, at least whilst under control of the practitioner. This also means that you could, in fact, transmute lead to gold, assuming you were adequately protected against the interesting varities of radiation that the intermediate states would produce. You wouldn't be able to create heavier elements this way... no transmuting lead into uranium, as gluing extra nucleons on needs the strong force (but if you have a wizard who can control that, maybe you could work together). Only transformations that conserve or reduce mass number would be allowed.
Note also that the weak force allows for the transmutation of a proton or neutron into its own antiparticle. This isn't elemental or isotopic transmutation, and as the resulting and inevitable annihilation with any nearby nucleons is not mediated by the weak force you'll get a load of quite unpleasant radiation in pretty short order. If you're not mkaing antihydrogen for CERN, this might only be useful as a sort of suicide bomb.
The practitioner would not be immune to the effects of radiation, as most damaging radioactive effects are mediated by electromagnetic interaction of ions or short-wavelength photons with matter. They may on the other hand be able to suppress radioactive decay in their body or immediate surroundings, making them much more resistant to the effects of radioactive fallout and you certainly wouldn't be able to pull a Litvinenko on them. They'd be able to benefit from cheap land prices around Fukushima, though they might not have many guests.
Given that depleted uranium is seen as something of a waste product of a nuclear-armed power's enrichment programs, turning this hazardous waste into valuable material might even be seen as something of a public service, and has the added benefit that uranium is so heavy that you could create almost any other useful material from it. Similarly, denaturing radioactive waste into shorter lived isotopes or even stable ones would be very valuable indeed.
Depending on how you want your powers to work, people with this ability might have some kind of material sense... you can't work out a nice decay path if you don't already know what collections of protons and neutrons you're dealing with, after all. How that turns into a useful power is up to you, but you can be sure that an alchemist knows their lead from their gold, and you'll not be able to sneak debased currency past them.
Killing people or damaging things by making them intensely and briefly radioactive or spontaneously fission seems a little crude in comparison.
Power-balancing this seems likely to be awkward, though restricting the range of the effect might be thematically appropriate. Don't accept massages from people wearing lead aprons and gauntlets, in case it wasn't already suspicious enough! Restricting the effects to elementally or isotopically pure materials might also work, so you might transmute a lead pipe but not a person.
The biggest problem is of course what to call them. "Weakling" would be an obvious epithet, especially from people capable to chucking lightning bolts or levitating, but probably best used only by those who are bored with the lack of cancer in their life.
I find it somewhat helpful to think of the Weak Force analogously to the Electromagnetic Force (there's a reason they were the easiest to unify, after all). Neutron decay can be modeled as a free neutron emiting a W- boson, which immediately turns into an electron / antineutrino pair. Compare to a charged particle emitting a photon, which, given high enough energy, could turn into an electron / positron pair.
So, just as it's necessary to ask what aspect of the force one manipulates and in what way for Gravity and EM, we must consider what specifically we're doing with the Weak Force, even if we ignore the more exotic things like extending the lifetime of Muons to start cold Fusion in the oceans and blow up the Earth (bwahahaha?). Messing with W or Z bosons directly would be analogous to messing with light, and I'm not sure what you'd be able to do with that (what is the mechanism for a carrier boson becoming leptons? I just plain don't know, and all that comes to mind is "something something Higgs Field", which is way beyond the scope of this question). So maybe you're controling when and how which particles emit Ws? In which case, you can make, say, 1/6 of the electrons in a piece of diamond give up their Weak charge, aiming the W- at the carbon nuclei to which they're bound. The electron becomes a neutrino, and if the W- hits a proton, it becomes a neutron. Congratulations: you just turned Carbon12 into Boron12. Or maybe you want a piece of Lead to spew the Weak charge from its neutrons until it becomes a piece of Gold.
Both examples mess with the chemistry, though. What would happen if all the Carbon12 in a diamond suddenly became Boron12? You just lost an electron, and the proton mandating its orbit. Is the diamond going to crumble? Most of the time something like this happens, it's in a plasma that just doesn't care, or in a neutron star, which also doesn't care. What if you're not converting a solid to a solid,? What if you're converting Carbon to Nitrogen? Either your diamond immediately sublimates, or you just found a very weird way to freeze things (now, explain where the temperature went, because Thermodynamics isn't giving up that easily).
Either way, the easiest way to kill someone with this power is to transmute the atoms in their body. One way or another, that's unhealthy. If you want to be flashier, bake a bucket of heavy water under a shower of long-lived muons. You might not have an easy time Weak-bending your way through steel, but wood, plastic, or any other solid that relies on light elements could probably be damaged quickly (what happens if you turn bound hydrogen atoms into neutrons?). And if you can vaporize diamonds by turning them to nitrogen, or turn NaCl to NeCl (or is it ArCl I always get Neon and Argon mixed up), that could be fun, too. You know, until you turn a bucket of salt into a bucket of neon and chlorine, which might be a war crime if abused.
