abecedarium would come from abcd, no relation to
dare.
elementa is sometimes thought to derive from the letters L, M, and N, though that's not the only theory.
What you're seeing here is something that will come up a lot with English->Latin dictionaries, namely that a lot of concepts which we have really strongly established terms for in English did not have terms in Latin that would make sense outside of specific contexts. This is why, beyond sentences in Latin composition textbooks that are carefully constructed to help you practice English->Latin skills, it is very difficult to compose good Latin with just a dictionary until you gain a good amount of reading experience and experience using a detailed Latin->English dictionary (plus word search tools like the
PHI Latin Texts website) in order to understand how the Romans would express a thought.
The term
abecedarium is only found in late Latin. It's probably the best term for the concept of the alphabet, though; different people will have different opinions about how "classically" one should compose Latin, so many would object to the use of
abecedarium on the grounds that it's not classical.
The word
elementum is classical and often used to mean the first rudiments of knowledge in a specific field, or of course the elements that make something up. Suetonius seems to use
elementa to mean something like "alphabet": "quartam elementorum litteram, id est D pro A et perinde reliquas commutet." (describing Caesar's cypher: "one should switch the fourth letter of the alphabet, that is D, for A, and likewise for the rest"), but that's highly context-dependent and
elementa would probably not be readily understood to mean "alphabet" outside of context. The phrase
prima litterarum elementa ("first rudiments of letters") comes up, but that refers to the knowledge of the alphabet and how to draw/pronounce the letters/spell words, rather than to the alphabet itself.
Since both of those terms are slightly problematic, Cassell uses a more periphrastic phrase derived from Quintilian: "Neque enim mihi illud saltem placet, quod fieri in plurimis video, ut litterarum nomina et contextum prius quam formas parvoli discant." ("Nor at any rate does this practice please me, which I see happening in many cases, that children learn
the names and order of the letters before their shapes"). Quintilian here is talking about a specific problem he has with early education where people memorize the alphabet in order but don't get good at recognizing the individual letters outside of contexts, so he recommends that teachers also present the alphabet in rearranged orders so that children are recognizing the letters from their actual shapes and not just knowing like "ok this must be C because the last letter was B". So in this context,
litterarum nomina et contextus ("the names and order of the letters") does refer to the alphabet, but simply giving that as a translation of "alphabet" ignores the fact that Quintilian uses such a long phrase because of the specific concept he's trying to emphasize, which is a contrast between rote memorization of "C comes after B" and actual knowledge of the letters in the form of "this shape is a C".