I'm curious if there is a formal name for the standard 7-die set (d4, d6, d8, d10, d10*10, d12, d20), and is there any backstory or history associated with it?
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The reason for the set mix as it exists is that, originally, the dice available were a set of platonic solids, sold by an educational company and repurposed by TSR. Namely, a tetrahedron (d4), cube (square hexahedron, d6), equilateral octohedron (d8), dodecahedron (d12), icosahedron (d20). This was a "platonic solids" set.
D20's were routinely read as d10's in wargaming, and used for generating percentiles, which made use of the d20 as a d10 or pair as a d100 a standard practice, going back as early as 1972.
A 1938 dodecahedron with 2 rounded faces opposite each other was available in 1938 for stock simulations. ➀ It wasn't used by TSR, but shows the probable origins of the pentagonal dipyramid we now think of as the d10.
As to names - the earliest modern set (with dipyramid d10) I can readily find documented by name is the "Dragon Dice" set from 1981 (as evidenced on the packaging copyright notice), as photographed at Dice Collector. ➁
I can, however, cite TSR "catalogue" from a ©1979 TSR product - Swords & Spells lists the older set as "Multi-Sided Dice Set" in the catalogue extract on the back flyleaf. ➂ It reads:
The same title and text was used in the 1975 product list in Strategic Review Vol 1. Issue 3 (1975). ➃
➂ Gygax, [E.] Gary, Swords & Spells, 6th printing, Tactical Studies Rules, 1979. Original copyright 1976.
➃ Tactical Studies Rules, Strategic Review, Vol. 1, No. 3, Autumn 1975, Ed. [E.] Gary Gygax. TSR advert on page 8. From the Dragon Archive CD.
There is no name for the full set other than "a set of polyhedral dice."
If I need to distinguish it from another set of polyhedral dice: I would say a Set of polyhedral dice suitable for playing DnD, as compared to a Set of dice for playing L5R (10d10) or a Set of dice suitable for playing Dilettante (10 d8s and 10 d4s)
The d4, d6, d8, d12, and d20 are the Platonic Solids. They are the only polyhedrons that have all sides the same shape and area, and have the same number of faces meeting at each corner. Platonic solids are ideal dice.
The d10 and the d% are not Platonic Solids. While the most common diamond d10 shape (pentagonal trapezohedron) does have faces the same size and shape, it does not have same number of faces meeting at each corner. (The top and bottom points have 5 faces meeting; the other corners have 3 faces meeting.)
d10s, while not being Platonic Solids, are super useful because of modern humanity's decimal thought process. (Base 10 number system, 10 fingers-plus-thumbs.) The two d10s for a d% is great because people are used to thinking of percentage probabilities, so can judge things well.
To get to the d10 diamond shape we know today actually took quite a lot of development.
I have an old d10 that looks more like a d12 but with some faces rounded off.
Calling them a full set of dice can be misleading, you can get a actual d100 (looks like a golf ball), as well as d16s and d32s that look like d10s . And a huge variety of other weird ones.
Wizards of the Coast published an article on the history of the dice. They offer no name for the set that was included in early DnD (and later), other than referring to them as "a full set of polyhedral dice."
The manufacturers (Chessex, Gamescience, etc.) just refer to them as 7 die sets.
The D4, D6, D8, D12 and D20 collectively are known as the Platonic Solids, however there is no commonly accepted name for a set of these dice plus the two percentile D10s other than a "Dice Set" or "Polyhedral Dice". See this Wikipedia entry for more information.
A term I have read, heard, and used in conversation (which to my mind indicates a fairly broad understanding in the community) is just, "seven-set."
As in, "Joe got a cool new seven-set with steampunk gears."
The common name I always knew them by is "Dragon Dice" but I think that may have been a reference to an old TSR game rather than the dice themselves.