Eric Clapton: Blackie, a black Strat pieced together from other strats. Brownie, a brown-burst Strat bought at a pawn shop.
Lucy, a '57 Les Paul gold top that was refinished red, later given to George Harrison. Clapton’s work on “While My Guitar Gently Weeps” was recorded with this guitar.
Jimmy Page, Led Zeppelin: #1, his favorite Les Paul, a 1959 acquired from Joe Walsh. #2, his other/second favorite Les Paul, also a '59
And there is the Frankentele used by Steve Morse. Telecaster body, Strat neck with multiple pickups. But for about 20 years he’s been using his own signature model built by Music Man Guitars to his specifications.
Billy Gibbon’s of ZZ Top has his 50’s Les Paul sunburst Pearly Gates
Albert King named his Flying V’s Lucy - he had an original 50’s and then a custom-made one.
John Mayer’s favorite Strat is one he helped make in the Fender Custom Shop - it is referred to as “The Black One” - to differentiate it from Clapton’s Blackie and Gilmour’s Black Strat, I guess.
Jerry Garcia had a few, but his most famous was the handmade Tiger - sold to Jim Irsay of the Colts and the subject of a ranty thread by me a year or two ago. Apparently it weighs more than a boat anchor and is not an easy-to-play guitar, but clearly Jerry made it work.
There are many, many players who are clearly associated with a signature guitar or two, but the guitars don’t have famous names and are just referred to by make and model. Or there is the guitar that Garry Moore (Thin Lizzy) bought from Peter Green (Mayall, Fleetwood Mac) that became known as the Peter Green Les Paul with the “Peter Green mod” (a magnet got flipped, so when both pickups are engaged, it’s tone was different vs. normal).
Just to muddle things up: with 50’s sunburst Les Pauls going up to hundreds of thousands of dollars, many have gotten names, kinda like Strad violins. Some, like Pearly Gates mentioned above, were named by their most famous owner. Others are named after a famous buyer, like the Stanley Burst, named after Paul Stanley of Kiss, who only had it for a few years before selling it. One of the most famous is the Brock 'burst, named after the vintage dealer who unearthed it and sold it. Along with the Stanley Burst (which is featured on the cover of the essential book on sunburst LP’s, Beauty of the Burst) - those two are considered to have the quintessential Sunbursts for vintage Les Paul…
Oh, and back to other guitars: Billie Joe of Green Day started off with Blue, his Japanese-made Stratocaster copy, but moved over to Floyd, a vintage Les Paul junior that was the basis for his Gibson signature model…
Peter Frampton’s customized black Les Paul, which he used on most of his early recordings (including “Frampton Comes Alive!”), was considered lost for decades after it went down in the crash of a cargo plane (along with the rest of Frampton’s gear) in Venezuela in 1981.
A fan discovered it several years ago (it was being played by a musician in Curacao), obtained it, and it was restored by Gibson before being returned to Frampton in late 2011. It still bears some burns on the body, and it’s now known as the Phoenix guitar.
Jeff Beck has had a few that have acquired nicknames that he uses - like the TeleGib, the Telecaster routed for Gibson humbuckers and a Gibson bridge and tailpiece that Seymour Duncan made and traded to him for the Yardbirds Esquire that the previous owner had sanded a body contour on (commonly nicknamed the “Esquire that Seymour got from me that I wish he’d give me back,” at least it seems that way by Beck. The TeleGib was used on Blow by Blow.
The Oxblood - the ‘54 stoptail Les Paul, routed for humbucking pickups and refinished from a goldtop to a dark reddish brown that photographs black a lot of the time. Featured on the cover of Blow by Blow. Was a custom mod done for a customer at the old Strings n’ Things guitar shop - they didn’t like it so it was up for sale when Beck was visiting.
Tina - the pink superstrat that Tina Turner carved her name into. Used on his 80’s stuff.
He has a favorite Strat - Olympic White, Rosewood fingerboard, custom wound pickups by John Suhr, etc., but I don’t know that it has a name…
ETA: as for the Neil Young Old Black reference. Of course! And what is it about Black guitars? More nicknames so far vs. Red or other colors…interesting.
There’s also it’s brother, “BumbleBee”, another Charvel hybrid pictured on the VH II album.
Sadly, Dimebag Darrell had always wanted Eddie to make him a replica but Eddie offered up the original to be placed in his casket after his murder.
And “The Shark”, which began as an Ibanez Destroyer but Eddie added 2 humbuckers, cut a ragged chunk out of the body and added some others details.
The “5150” was similar to the Frankenstrat but began as a Kramer Pacer.
The OP lists Jimmy Page’s Les Pauls, but there’s also his Dragon Tele - the one he got from Jeff Beck and used on LZ 1 and 2 before some “friend” refinished it and ruined it.
Roy Buchanan’s '53 Tele was Nancy.
Observation: in guitar circles, on guitar message boards, there is a regular thread started that basically asks “Do you name your guitars?” The VAST majority go out of their way to say that: a) they do NOT - only shorthand names based on make, model and/or color; and b) naming your guitars is silly, bad and uncool.
Just interesting that so many Heroes name theirs. Sure, many are based on make, model and color (Black, apparently ;)), but “Mighty Mouse”?
As I have said in previous threads, naming a guitar is a tricky thing. I’ve tried hanging names on many guitars that really didn’t take - I quickly revert to make/model/color. Occasionally, a name gets attached to a guitar early in my acquiring/building it and I never end up calling it anything else.