What does a tonearm contribute to the sound of a turntable?

Fluid damping causes all kinds of problems of it's own. The solution is worse than the cure. 
I went from various iterations of the original SME 3009 arm to various VPI arms including two of their 3D arms to my current two arms, Reed 3P's. 
I have had 9", 10.5, and 12. Steel, alloy, carbon, plastic (VPI 3D, call it "composite" if you like, it's junk IMHO), and now one Reed is 12" cocobolo and one is 10.5 in ebony. 
Just my opinion and experience-everything about the arm makes a difference. How it is tightened, how the vertical and horizontal motion is designed, the materials, the length and materials used for the armwand, shell, clips, wiring, hell, even the mounting and armboard materials have an effect on overall sound. 
Many of these things are not worth sweating about and some are. But they all have an effect on the overall sound. 
I use the same old analogy often but it is a good one. A pro can take a better photo with his phone than you can with a 10K Nikon set up. But that doesn't mean that the 10K Nikon set up is not worth the money in the right hands. Likewise, you can get great sound from a modest set-up if you set it up optimally. But.....

@fsonicsmith Cool. Thanks for the clarification. Another thing I've often wondered is how good the tiny wires connecting the cartridge to the tonearm are. With all the fuss and expense about IC's and speaker cables, are these overlooked?

There is really not that much mystery involved with tonearm wire. The same concept applies-basic tonearm wire is fine with basic gear and as you go up the chain, the quality of the tonearm wire makes a subtle difference and improvement. Btw, "tiny" is endemic to tonearm wire due to the very low level signal. All that said, I am a true believer in FireWire AG, a wire that is manufactured in Germany by  Stereo-Lux Musikanlagen. Something has to account for the amazingly transparent sound I am getting with my Reed arms and Reed uses this wire. It is incredibly pliable as opposed to being stiff, with a very soft clear rubber-like insulator. The clips are unusually nicely machined, they just exude quality from end to end. 

 It takes special test gear to measure the resistance between good connectors, normal test meters won't pick it up.

Do we have to go down that well-worn rutted road again?

@fsonicsmith welcome to the club, i have the same Reed 3p Cocobolo "12 :)) Wonderful modern High-End tonearm. I've met people from Reed in Vilnius, Lithuania to get my sample. It's definitely a top quality tonearm. Then i went to vintage route and it's an interesting journey with tonearms like FR-64fx and 64s, Lustere GTS-801, Technics EPA-100 MK2, Ikeda IT-345  

It was you Chakster who caused me to look into Reed. 

@fsonicsmith Wait - I'm curious. What well-worn, rutted road?

With the new Jelco, I now have not one, not two, but six signal interfaces between the cartridge and the speakers:

1. cartridge to headshell/arm
2. headshell/arm to Premiere Interface DIN (little box on the back of the VPI - which I desperately want to get rid of)
   
3. Premier Interface DIN to RCA
4. RCA to phono pre
5. Phono pre to amp
6. Amp to speakers

Is this normal?

Well. let me set you at ease first. You need not worry about 5 and 6 because they are higher level signals. 
It is the low level signal that is most subject to being degraded by connections along the way. Ideally, the cartridge would come with integral wiring from cartridge to phono stage input or to go to the even more absurd (and nearly impossible) the cartridge would be hardwired to the circuitry of the phono stage. 
Connections = convenience. A removable headshell is chief among them. 
Is it worth sweating about? Absolutely not. Only at the Mike Fremer level or the Whats Best Forum level do the connections likely make an audible difference. This is just one man's opinion-my own. 
My "problem" is with a reference to testing. The argument that if it can't be measured it does not count is as old as rocks and persuading the camp that adhere to that argument that they are misguided is like trying to move the Rock of Gibraltar. 
And yes, I agree that the VPI junction box is something to "desperately want to get rid of". I hated how it got in the way of trying to use the conventional anti-skate mechanism. Hell, I hated everything about it. 

lewm7,760 posts04-28-2020 5:00pmThe wire is probably fine, but Reed tonearms have a lot more going for them besides good wire. Otherwise, we could all buy FireWire and use it to re-wire some el cheapo tonearm. (Another owner of a Reed tonearm with cryo-treated FireWire here.)

I could not agree more.