Listen to Larry Carlton, Jeff Beck,Steve Morse, Grant Green, Freddie/BB/ King, Hound Dog Taylor, Luther Allison, Dave Gregory ect.. - love'm or hate'm they all have very distinctive tones, Why? It's not all about the amp and the effects - what else gives a guitarist his/her tone?
It's in their fingers, by which I mean the way they attack the string, the way they use vibrato, the inflections they use. All of these things contribute to tone in subtle ways.
Picking: Where and how you pick makes a huge difference: picking close to the bridge makes the guitar sound bright and sharp, closer to fingerboard makes it sound rounder and warmer. Jaco Pastorius got a lot of his distinctive bass sound by picking really close to the bridge, thus accentuating the upper harmonics and making the tone more nasal and bright, but then turning the tone knob all the way down to round out the sound.
Using heavier or lighter picks, fingerpicking, tapping -- these all contribute as well. Charlie Hunter, for example, uses the fleshy parts of his fingers rather than the tips with the nails, because he wants that rounder sound. Wes Montgomery used his thumb and got a huge, round sound. I myself use a very stiff pick instead of a lighter pick, because the heavier one aids in note definition; when I fingerpick, I always use my nails for clarity, but never metal fingerpicks, which have (to my ear) an unpleasant, scratchy sound.
Vibrato: As for vibrato, well, it's hard to be too specific, but I'll give it a shot. When I think of distinctive vibratos, I think of Angus Young's narrow, fast tremor, B.B. King's subtle waggle, and Dimebag Darrell's huge, wide shake that just sounds so deliciously evil (it almost seems as if he's practically bending a half-step). Each of these players' vibrato was/is so distinctive that I think if I heard them playing any guitar, I'd recognize them.
You can, of course, practice vibrato and work on getting a particular sound and style, but in interviews with players who have distinctive vibratos, when asked how they developed it, they almost all say something like, "It just felt right/natural." This makes sense to me: if you do it the way someone else teaches you, for better or for worse it's less likely to sound like you and more likely to sound like the-way-you're-supposed-to-do-it.
Inflection: By inflection, I mean the subtle aspects of playing a note that aren't it's specific pitch and duration. Vibrato is a part of that, for sure, but so is whether or not the player bends into or out of the note, slides up to or down to the note, plays grace notes, gets to the note via hammer-ons or pull-offs or tapping or some other way, or any of the myriad other ways to get the notes to sound.
By way of an example: Clapton's solo on "Sunshine of Your Love" is like a case study on bending as a type of inflection. Almost none of the notes are played straight; he either bends up to the note or releases the bend down to the note. The tone is already highly compressed and smooth due to the extreme distortion, but the bending increases the lyrical quality of the sound and contributes to its vocal-like tone.
What influences a player's tone has a lot to do with the equipment they use. A Strat sounds slightly different to a Tele, and a Les Paul sounds drastically different to a Rickenbacker.
The construction of the guitars and the pickup used has a big effect on this. A semi-hollowbody guitar on the neck pickup is going to sound worlds apart from a solid body guitar on the bridge pickup.
The wood used alters the sound of the guitar. For example, borrowing what I posted in response to an earlier question:
The amp and effects that the guitar is played through also makes a massive difference. Each different amp and effect colours the sound differently, depending on whether it is valve or solid-state, size of the speaker, the EQ settings, built-in effects, it goes on forever.
The technique that the player employs can also make a big difference. Obviously the sound is going to be louder and harsher if you smash the heck out of the strings with a plectrum than if you gently fingerpick open chords. But the way you pick makes a big difference, along with the way you actually manipulate the neck. For example, The Edge of U2 uses a plectrum to pick out doublestops, and to play chords that he has selected or created variations for specifically to sound open and ringing as possible. The result is a chiming, sustained and very open sound. A player that wants to sound very heavy and rhythmic will employ power chords, whereas another player may choose to fingerpick out melody lines.
Strings and plectrums also make a difference. Heavier strings give a much thicker tone that has more presence than lighter strings. I have found that lighter picks are easier to play quicker with, allowing melodies and notes with more sustain, to be picked out (see Johnny Marr of the Smiths), whereas a heavier pick is better for rhythm and power chords. That may not be true for everyone, but that is what I have found in my own experience.
Hope this helps.
Equipment plays a significant role in a guitarists tone, as does technique, but another important factor to consider is post-production. For example, if I take a pair of Metalcore bands, hand them all PRS SE 245's and a Boogie Triple Rec and tell them to go nuts they will sound very close to alike. They won't sound exact due to inflection and expressiveness in each players technique, however in post production these two bands could be engineered to sound very distinct. It happens every day.
My shining example for this is Tom Morello. When he was in Rage, he had a very distinct tone. I believe early on he played a Telecaster with active pickups and a killswitch through a Marshall stack plus he banged around on a Digitech Whammy a lot--but his trademark tone is likely the union of his unorthodox playing technique and his gear. I wouldn't necessarily weight one more than the other. If you listen to AudioSlave you will still know it's Tom Morello based on his technique and tone but it sounds remarkably different than early Rage--more organic and warm. That's post production. It still walks like Tom Morello, talks like Tom Morello, and swims like Tom Morello, so it must be Tom Morello. Advances in recording equipment and technique are also factors here.
So in essence, I believe artists work hard to come up with a sound that they are happy with based on technique and gear. The factors in "good tone" are indeed endless (and subjective), and gear plays a significant part, but truly innovative tone is usually tightly coupled to technique. The sum of the parts are not weighted equally in every situation mind you. Eric Johnson has a trademark sound that is built specifically from his gear--but there's no way I could emulate his playing even if I had an exact copy of his rig. My fingers just don't bend that way. In this situation his technique drives his tone, not vice versa.
Additionally, artists aren't known to be the most affluent people in society so most of them start out on economical gear--another point in favor of technique. Early slave blues pioneers certainly didn't have the money to purchase nice guitars but I would definitely argue that they defined the blues genre via their technique. You can't argue with the tone of Lucille, but she just wouldn't sound the same without the King at the helm.
A lot of it is in the effects, amps, guitars, etc... You can find this out if you ever get to see them play on low quality instruments. BUT a lot has to do with their style, how the play phrases, note choice, their confidence, etc... You can see this when they play a simple scale then play something more interesting.
Without a good sound producing device it doesn't matter how good you are you'll suck. Think of a piano that is completely out of tune(or no sound at all)... you think people will pay to see that? But at some point "talent" takes over and what your hearing is experience.
People that say that "it's all in the hands" have no clue and they themselves are not any good. No good musician will say that(how many have you heard?). They know that good sound producing instruments are a must(that's why they spend so much money on them) but also a lot of practice.
Problem is what most people consider as "tone" is not tone at all as they are confusing "music" with tone. Tone is purely the sound. A musician has little control over that through their hands(unless you consider handing over money to buy better equipment as "in the hands"). How hard they pick, hand position, finger strength, etc all play a minor role in tonal production. e.g., changing your pick can be a big change in sound and feel for you but your average listener isn't going to know the difference.
What it is, is a lot of small variables that all add up to a significant portion to tone. They have gotten this through luck, practice, talent, and experience. But it's only about 10% of the tone if that(more like 5%).
What most people "hear" as tone is really the musicianship of these people. Just study the tone of these guys when they play something simple and you will see it's not much different than anyone elses(must compare similar tone). There are big differences but are so subjective as not to be comparable.
It's all about how they play that makes you think they have good "tone". I have see some great musicians have sorry tone but "sound good". It's because they had so much musicianship they could "mask" the bad tone(not intentionally, they were just good). Even then it's hard to tell they have bad tone until they play simple enough stuff so you can focus on the tone itself(and not how they are manipulating it).
You can still "sound good" on crappy gear since most gear now days is good enough. A cheap ephiphone will get you 50%(more like 75%) of the sound of a 10k les paul quite easily. If you have a lot of talent you can "sound better" than a guy playing on that les paul... but if you played on the les paul you will sound even better.
If you are simply wondering why these guys are good and make good music it has to do with all the factors. They have it all... luck/determination is a significant part of it. There are a lot of guys in the world just as good or better than have absolutely no notoriety.
I'm sure you've had the experience of listening to these guys at some point and asking why there tone sucked? If not you haven't see them in impromptu situations enough(I haven't for most but have for some and doubt it's any different).
Hand any one of these guys a guitar that simply will not stay in tune and is out pretty bad and if they actually stay and play it you won't say they sound great... and if you didn't get to know who it was you would say their tone sucks.
I think technique trumps equipment. Metheny sounds like Metheny regardless of what he's playing.
I don't think any beginner should spend much time thinking about tone. The very best players I've jammed with or done workshops with say the same thing: Do what sounds cool. If you try to 'cultivate' a tone, you're doing it wrong. It's just a natural result of skill maturity.
We all know good equipment is a must. But you can give the shiniest stack of gear to a weak player and it sounds ghastly.
It's kind of like 'what pick should I use?'. Answer: Whatever feels good and sounds good.
It's already been hit-on, but to really get who has a good command of their tone, listen to them on an acoustic if they are primarily an electric player. Or nylon.
The best example I can think of is Chet Atkins. It's not what he plays, it's 'how' he plays no matter the equipment. One can hear Chet playing even when he is not playing in his trademark fingerstyle (sadly referred to as "boom chick").