I have been researching how to create a simple water conductivity sensor. The easiest way I have found involves putting two wires (electrodes) into the water and measuring resistance/voltage between them. However, all the projects I've looked at use additional wire, not jumper wires directly, as electrodes (for example, here and here). My question is, would it be possible to simply use jumper wires I already have (assuming I can ensure they don't touch in the water)? If I don't want to buy additional copper/nichrome wire and put Arduino jumper wires into water as electrodes, would that work? Thank you.
No.
Bare copper wire OR better yet tinned copper wire will work as electrodes. The issue is the copper will start to react with the water and can quickly disappear. For short tests its probably OK but for long term could be a problem.
Question do you happen to have any thermocouple wire?
OR
tin lead solder (solid w/o flux will be best).
Both could be used.
The "trick" is to avoid electrolysis and probe corrosion. The customary methodology is to swap potentials (low frequency polarity inversion: like AC) but corrosion is difficult to prevent with tin-plating on iron pins (typical jumper pin leads.)
If you just want to play, sacrifice a couple of jumpers.
You can also look into how projects doing plant-needs-watering are implemented as some of the write-ups go into detail about corrosion of electrides.
So we have that sorted. 
Various possibilities include the carbon rods out of old non-alkaline batteries, nicely cleaned and with wires soldered to the metal caps, then the cap and the wire join waterproofed with silicone.
Electrides?
Basically, if you want it to work for more than a day or two, you use the "capacitive" sensor modules which are (or should be if correctly manufactured) fully insulated and thus waterproofed.
Thank you for your response! Could you please elaborate on it a little bit please? I am relatively new to Arduino (and to measuring conductivity in particular), so I would really appreciate if you could explain some theory behind it.
Unless steam, evaporated water, sensing sensors.....
Measuring conductivity with a DC voltage/current source and two metal wires in a conductive solution is essentially the same as Plating.
When Plating metals, i.e. chome on steel (simplified) the source metal and receiving metal is put in an electrolyte solution. A current is passed through one metal -> electrolyet -> receiving metal.
The process causes some molecules from the source metal to go into solution then travel to the other metal (receiving metal) and gets deposited.
Yep, one option. Another common method:
https://www.sparkfun.com/products/13322
One commonly known issue with soil moisture senors is their short lifespan when exposed to a moist environment. To combat this, we’ve had the PCB coated in Gold Finishing (ENIG or Electroless Nickel Immersion Gold).
And the elaborate, commercial thingies:
https://extension.umn.edu/irrigation/soil-moisture-sensors-irrigation-scheduling
The project is for Science Olympiad, so I am unfortunately not allowed to use any pre-manufactured probes or sensor modules. If I put the wires into a plastic straw and then use hot glue as sealant to make it waterproof (so that the the only parts actually touching the water are the non-insulated ends), could that work, or would it corrode too quickly? And if I do use additional wire, what material would be best - copper, tinned copper, nichrome, stainless steel, titanium, or something else?
I don't believe Carbon plates, but it has some resistance. Perhaps going to the hardware store and purchase two of a decent size motor brushes. They already have wires on them. Insulate the wires from the water and try them out.
You never mentioned how you plan on measuring the resistance with the Arduino. Do you have anything specific planned.
What type of things might make this project standout?
Why not just make a bunch of electrodes, swap 'em every day or so before demo. That way you can be assured of the same results.
Home improvement stores and hobby shops often have a selection of wires and rods. Stainless would work well and instead of changing, just clean it with a scrub pad.
Thank you, that should work great! One more question - do I have to insulate the electrodes?
Electrodes do not require insulation. A bit of colored heat shrink tubing at the top may give it a bit of flair though. 
You can use alligator clips to attach wires to the electrodes. The voltage is so low, it is of no hazard.
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If you insulate the wire, it cannot measure the conductivity of the liquid.
I know this, I meant insulating a part of it. Essentially leaving an uninsulated end that will actually measure conductivity but insulating the rest of the wire (like in the examples I put in the original post). Would that be required?
Well you want to have ALL the conductive wire in the liquid, else the conductivity will change because the measurement area will change. So yes you should insulate some portion of the wire. You should also make the wire area large to get a better signal.
Now be careful of:
- liquid sneaking up your insulation resulting in a measurement offset.
- make sure your support structure does not collect moisture. A pencil as shown in one of your links is a horrible suggestion. Pick some hard plastic.
Oh well, I tried. 
I saw your post and although I didn't comment but I was curious if you knew of a relationship between conductivity and capacitance?
The capacitive probe is essentially assessing impedance, and assuming the capacity - of the insulating coating on the electrodes - is more-or-less fixed.
My exasperation was in respect not of the capacitive probe suggestion, but my more practical suggestion for constructing electrodes.