History

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.

Names

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:

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

So:
Multi-Sided Dice Set - the original d20 d12 d8 d6 d4 platonic set.
Dragon Dice - the "mud dice" set with d10's.

➀ Dice Collector - Mason & Co Stock Exchange Dice - www.dicecollector.com/MINT39_MASON_&_CO_STOCK_EXCHANGE_DICE.jpg

➁ Dice Collector - TSR - www.dicecollector.com/THE_DICE_THEME_TSR.html

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.

Short answer: really, practice is the only answer

It's like any kind of memorization task, eventually you're going to get it, and you'll have trouble until then. But there are ways to make the memorization easier.

1. You are going to have to correct them sometimes. Don't think of that as a failure. Just make the correction and move on.
2. Don't default to telling the player what to roll. People tend to learn better when they make a mistake and get corrected than when they're following rote orders. So if they're supposed to roll a d20, let them grab a d12, and let them take a second to ask you if it's the right one or not. Depending on the player's style, maybe just tell them "yes" or "no," rather than telling them which die.
3. Designate one person to help. At a lot of tables, every. single. person. tries to help and it can be really overwhelming for a new person. When you have four people all trying to help louder and faster than one another, it can sound an awful lot like four people screaming at you for getting something wrong. Maybe the designated coach is the DM, maybe it's a player (if the new player was brought in or is close friends with a more experienced person, they're a great choice for this), but tell the group as a whole to simmer down.
4. Be patient, and let them know it's OK. A lot of times, you're totally OK with a bit of a delay and are accommodating the new player, but they don't know it. Sometimes a new player will decide to drop out if they feel like they're being a burden to their friends, and that's not cool.

In terms of being able to distinguish them, a good mnemonic can be to have the player set the dice up in front of them with the highest number facing up. That's a quick numerical identifier and it lets them reference the correlation between the shape and number while they aren't rolling so they can be quicker to act when they are rolling.