There are a number of possibilities for ways that speaker wire could affect sound.
Larger wire will reduce DC resistance. Larger wire that isn't stranded will increase inductance, which will lead to increased impedance at higher frequencies. The insulation used will affect the capacitance between the wires. The construction of the cable (e.g., straight zip wire versus twisted or braided wires) will affect the degree to which outside electrical signals will induce a signal in the wire.
To get at least some idea of how much affect these factors could have, ran a few signals through some wire, and into the oscilloscope. The first piece of wire was a fairly normal 6 foot (or so) zip cord -- a bit heavier than many (16 gauge) but nothing terribly special. [Edit: Here's a picture: ]
The second was a piece of "special" speaker wire the store threw in when I bought an amplifier many years ago. If memory serves, it's called Kimber cable. It has 8 separate wires (4 of each color) that look like they're probably around 20 gauge each, all braided together:
I then took the 1 KHz square wave signal my oscilloscope supplies for calibrating, ran it through each piece of wire, and captured the result:
The upper trace went through the Kimber cable, the lower through the zip cord. As is pretty easily visible, there's clearly more noise present on the signal that went through the zip cord.
Now, I should point out that this is only a 400 mV signal, and the noise on it is peaking at about 150 mV. That means the power it represents is quite small: P = E2/R, and it's driving a 10 megohm oscilloscope input, giving roughly 2 1/4 nanowatts. If, for example, you were listening to music using an average of 1 watt, that would be a noise level around 86 dB down. Caveat: this is measuring well up into the radio frequency range, so a loudspeaker may not be able to reproduce all the noise being measured here -- but most of it is almost certainly at 60 Hz, which most loudspeakers can reproduce.
OTOH, this is noise induced after the amplifier, so it will remain nearly constant regardless of how loudly or softly you have the volume set. I'd have some serious doubts about being able to hear noise that's 86 dB down from the signal, but if you turn the volume down a lot, this would remain (at least close to) constant, and I can believe that it might start to become audible.
For reference, here's a picture from just touching the scope probe with my finger, but not the ground:
The basic sine wave is at 60 Hz, and the "roughness" of it represents the signals at other frequencies. Obviously, the vast majority is at 60 Hz.
I also did a trace of the 1 KHz square wave going directly from the test output to the 'scope's probe. Here it is:
At least offhand, I can't see enough difference between the direction connection and the Kimber cable to claim I'm measuring any difference at all. Perhaps tests at higher frequencies or using different signals, etc. would show some difference, but at least based on what I've seen in this test, I can't claim to measure anything.
The difference between zip cord and "real" speaker wire is measurable, and right at the point that it's difficult to say with certainty whether it's really audible or not. The "real" speaker wire I used, however, is passing the signal through cleanly enough that I'm extremely doubtful that anything else could pass the signal much more cleanly (though perhaps a lot longer cable would degrade the signal enough to leave room for something else to be better). I don't know what the Kimber cable cost, but since they threw it in with an amplifier that cost something like $600 (IIRC) I'd be quite surprised if it was more than $15-20 or so -- certainly not even close to thousands of dollars in any case.
For what it's worth, the scope is a Tektronix 7854, and I used a 7A26 vertical input amplifier and 7B85 horizontal unit. Realistically, the 'scope probably doesn't make much difference though. It would be a lot harder to duplicate the exact wire I used -- I can't find a brand marking on it; all it says is 60o, UL/CSA approved, 16 AWG. For that matter, the room your stereo is in probably won't have exactly the same level of RFI as my office -- but I don't have any particular reason to believe my office is particularly above or below average either.