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For example, what is the "s" in "apples" or the "ies" in "ponies" called? I found that "morpheme" is probably close, since that's just the smallest grammatical unit in a language, but that's not really identifying the pluralizing nature of the morpheme itself.
I've suggested "pluralizer" to a friend, but that kinda seems made up and I was wondering if there is an actual grammatical term.
This is usually just called the "plural marker" or the "plural morpheme". In English it's usually "-s" though there are plenty of exceptions, and other languages of course have their own plural markers.
Pluralizer
https://en.oxforddictionaries.com/definition/pluralizer
Grammar. A suffix, prefix, inflection, or auxiliary word which forms a plural.
Your original suggestion to your friend is totally sufficient, as it turns out. It does feel slightly made up!
You may not be asking the question you think you are. Pluralizers always make something plural, and the examples you give -- apples, ponies -- do indeed make the words plural. But consider he goes/they go; the -es marks singular, but what marks go as plural?
There are different morphemes (the smallest chunks of sound-meaning correspondence, like un-, lock, and -able in unlockable, or book, keep, -er, and -s in bookkeepers) for nouns and verbs. As you can see, pluralizing verbs in English is a matter of a missing suffix, while singularizing them adds a suffix. And only in the present tense, note.
This is the opposite of the use with nouns -- the noun plural suffix inflectional morpheme {-Z₁} is added to the noun, which is otherwise singular. The third person singular present tense verb inflectional suffix {-Z₂}, however, gets added to the singular (he goes), and not to the plural (they go).
It's a nice little ironic feature of English morphology that, while there are only 9 inflectional suffixes left in English, three of them are completely identical in shape (which is why they all have the same morphophonemic shape -Z):
{-Z₁} (1 row, 2 rows) {-Z₂} (They go, He goes) {-Z₃} (That's Joe, That's Joe's) All the -Z's have the same allomorphs in the same distribution, like the famous 3 identical German personal pronouns sie 'she', sie 'they', and Sie 'you (pol)'. They're all pronounced Z /zi/, but they don't have the same syntax.
So, with regard to nouns, the final -Z (bats with /-s/ after voiceless, eggs with /-z/ after voiced, and churches with /-əz/ after sibilants) is indeed a pluralizer. But with respect to verbs, -Z isn't; it's a person/tense/number marker -- 3rd person, present tense, singular number -- whereas the noun suffix is just a number marker for plural.
Probly the best general term for the suffix on apples and ponies (both end in voiced sounds, /l/ and /i/, so they're both the /-z/ allomorph) is plural suffix. If you want to be formal, English Noun Plural Suffix.
This can also be called a "declension". A declension is an instance where the spelling or sound of a noun changes in order to give information about its number, gender, or grammatical case. While some languages have many noun cases (Russian, Finnish, and Estonian, for example), English has only two - singular and plural, so we rarely talk about them in terms like "declension" or "case".
In German, for example, which has four noun cases, the word "country" is "das Land", the word "the countries" would be "die Länder" and the phrase "in the countries" would be "in den Ländern". Each of these changes is a declension - first giving information about number, then about position.
While English doesn't have this level of grammatical complexity with nouns, the principle is still the same - a noun is changing its spelling and sound in order to convey more information about that object. Thus, a declension.
Any small linguistic element that modifies the meaning of a word is called a phoneme. The phoneme that makes a singular noun into a plural noun does not as far as I know have a particular name. If you invent one you might consider inventing a word to designate "ing" and ed". I doubt there is much use for any such word.
