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?
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 trapezohedron 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:
Multi-Sided Dice Sets — Each set contains one 20-, 12-, 8-, 6-, and 4-sided die
The same title and text was used in the 1975 product list in Strategic Review Vol 1. Issue 3 (1975). ➃
Multi-Sided Dice Set - the original d20 d12 d8 d6 d4 platonic set.
Dragon Dice - the "mud dice" set with d10's.
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.
Original D&D had no 10 sided die, and instead had a d20 number 0–9 twice. The modern pentagonal trapezohedron (diamond like profile) shape was developed in 1980 according to this history. Its benefit over the double numbered icosahedron (d20 shape) is that is is easy to tell apart from the normal 20.
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."
For what it is worth - the Blue covered "Basic D&D" Box in 1978 (mine was a fourth printing containing B1 - In Search of the Unknown) contained the following dice:
1d4 - Yellow 1d6 - Orange 1d8 - Green 1d12 - Blue 1d20 - White 1d20 - Pinkish Red
The d20s were both numbered with 2 sets of 1-0 numbers and you had to color one set into to make them work as D20's.
But it also allowed you to use the two dice to make percentile rolls.
Sets that do not contain two d20's are probably either incomplete or may have been produced in 1979 during the beginning or ending of the dice shortage that led to the use of paper chits and provision of an ordering coupon.
None of the sets currently (December 19, 2018) available for sale on ebay contain both.
They are commonly called Mud Dice due to the soft plastic used in their construction.