I can give you what the conscious inspirations were, assuming I won't just forget something (it was a lot of years ago now).
There were things that the initial design specifically and intentionally took from Gamma World, as someone has already mentioned, as well as the Player's Options (mostly from Combat and Tactics, like Attacks of Opportunity and some of the combat manuevers). Obviously, 1st Edition AD&D gave us the kernels for the assassin, the half-orc, the barbarian, etc., although it was more the concepts in that case than the mechanics. The biggest influence on 3E was simply D&D and AD&D.
Skills were obviously the children of NonWeapon Proficiencies. Feats actually came out of them as well. What we did was originally create three skill lists rather than one. We sat down and divided all the NWP and our new skill ideas into three categories by theme. There were "background" skills, like cooking, sailing, weaponsmithing, etc., and then there were "adventuring" skills like climb, hide, use rope, etc. Then there was the third category of skills (whose name I can't remember), but they were distinguished by the fact that you didn't make a skill check with them. They just gave you something new to do. Originally, each level a class got to pick a couple ranks from background, a couple from adventuring, and every third level got to pick a new one from the third category. Well, obviously the third category became feats (originally called "Heroic Feats"), and the first two got melded back together and class skill lists were created. (I must admit, there's a part of me that still thinks the menu option was kind of cool, though.)
As for Rolemaster, Runequest, GURPS, and so on, I always get a kick out of hearing what people think we based the game on (usually, it's their favorite game, so I'll take it as a compliment). Jonathan loved Runequest, and that influenced Ars Magica as well as his own D&D house rules. He likes to give credit where it's due to RQ, which is fair, but really that influence comes as much from his D&D house rules (which were influenced by RQ) than anything else. And as Jonathan has been quoted saying here, a lot of the things he wanted to introduce from that material when he joined the team we were already doing anyway.
In fact, I remember having discussions of changes where we went the other way and said "we could do that, but that's TOO much like Ars Magica." It was important to us for this to stay D&D.
Truthfully, though, other than Jonathan talking about Runequest and Ars Magica, I don't remember a single specific reference to another individual game system, as in "let's do this the way GURPS does it." Instead, what we'd usually say is "lots of games handle this issue in this way." Because as someone else said in this thread, the game industry is very incestuous. The mere fact that you can look at 3E and one person can say it's just like Rolemaster and another can say it's just like GURPS suggests that game design (particularly by the time we started design in 1997) had experienced a lot of cross-pollenization. So, knowing that 3E was going to be more skill focused than previous editions led us into discussions of "well, some skill based games handle this situation like this, while others do this." The three of us, particularly Jonathan and I, had a wide familiarity with a lot of games, including a lot of fairly obscure ones. What we saw were general conceptual paths that a lot of designers took on a lot of games. In some cases, we followed, and in some cases we tried to develop a new path.
As for Alternity, there were discussions about using some of the Alternity ideas in the game, but for the most part they were rejected. Many of them ended up in Star Wars, though.
Dragon Fist was really kind of the first d20 product, well before the concept of that was born. It took a lot of the ideas we'd been developing for 3E and ran with them in cool and interesting ways. Those ways were very specific for its genre, but it started the idea that the game system we were developing could be shaped for different genres and different purposes even back then.
There are also direct influences that are pretty much invisible, because they come from our own games. All three of us played a lot of D&D, and had developed either house rules or simply areas of the game that needed more development. The changes to halflings, for example, came out of the game I was running at the time, where they were far less like hobbits.
Prestige classes were, more than anything, a response to kits from 2E. When I started working on the DMG, lots of people asked, "so will there be kits in this book?" I didn't like kits, but I recognized their value to the game (in customizing your character) and so I focused on why I didn't like them. Primarily, it was because it required you to customize your character from the get-go before you got a chance to do any development and you could never go back. I had this idea then, to offer a way to "add on" customization later if you followed a certain path and I remember very vividly the conversation Jonathan and I had where we realized the new multiclassing rules would be the way to go. It's interesting for me to see the replacement level idea that WotC is putting in some of their recent products, because that's a bit closer to the original idea, actually. (Not that I regret anything about the Prestige Class mechanic--although I do remember wondering if the idea would catch on, and whether the five I created for the DMG would be the only five that ever existed. Irony.)
All that said, the basis of d20 sort of designed itself. Once we'd developed the "core mechanic" of rolling a d20 and adding some modifier to get at a target number (which is like a lot of games, admittedly), most of the rules just came as a natural development with us making each situation work as it applied to that core mechanic.
Our goal was never innovation. It was in many ways the opposite. To take an already existing game and make it more fun and more playable but without changing its feel. We never said "lets create a whole new subsystem for this aspect of the game that will be something no one's ever seen before." We said, "how can we make this situation play well with it still being D&D."
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