Learn Hardcore Portuguese (Portugal): Eu prefiro a de algodão.

adds emphasis (“I prefer the cotton one”)
can make the sentence clearer in isolation or in contrast (e.g. Eu prefiro a de algodão, mas ele prefere a de lã – I prefer the cotton one, but he prefers the wool one).

Why is it prefiro and not something like prefere?

Preferir (to prefer) is irregular in the present tense. In European Portuguese:

Eu prefiro – I prefer
Tu preferes – you prefer (singular, informal)
Ele / Ela / Você prefere – he / she / you (formal) prefers
Nós preferimos – we prefer
Vocês preferem – you (plural) prefer
Eles / Elas preferem – they prefer

In Eu prefiro a de algodão, prefiro is the 1st person singular form, agreeing with eu (I).

What exactly does the a in a de algodão mean here?

In this sentence, a is the feminine singular definite article (a = the), but it’s used almost like “the one” in English.

Contextually, it stands for a feminine noun mentioned earlier, for example:

Eu prefiro a camisa de algodão. – I prefer the cotton shirt.
When the noun camisa is clear from context, it’s dropped, and you just keep the article:
- Eu prefiro a de algodão. – I prefer the cotton one.

So a = “the (feminine one)”, referring back to something like camisa, saia, blusa, etc.

Why is a feminine? Can it change?

Yes, it changes to match the gender and number of the omitted noun:

Eu prefiro a de algodão. – I prefer the cotton one. (feminine singular: a camisa, a saia…)
Eu prefiro o de algodão. – I prefer the cotton one. (masculine singular: o casaco, o chapéu…)
Eu prefiro as de algodão. – I prefer the cotton ones. (feminine plural)
Eu prefiro os de algodão. – I prefer the cotton ones. (masculine plural)

So a is feminine singular because the understood noun is feminine singular.

What is being left out after a? What noun is missing?

There’s an elliptical noun – a word that’s understood but not said. For example:

Full version: Eu prefiro a camisa de algodão. – I prefer the cotton shirt.
Elliptical version: Eu prefiro a de algodão. – I prefer the cotton one.

Context could also be:

Qual é que preferes, a saia de lã ou a de algodão?
– Which one do you prefer, the wool skirt or the cotton one?

The second saia is omitted to avoid repeating it; a stands in for a saia.

Why is it de algodão and not just algodão?

The preposition de often indicates the material something is made of:

de algodão – made of cotton
de lã – made of wool
de couro – made of leather
de madeira – made of wood

So a de algodão literally means the [feminine one] of cotton, i.e. the cotton one.

Without de, algodão is just cotton as a substance. For this construction (describing the material of a product), de is required.

Why isn’t there an article with algodão? Why not do algodão or do algodão?

Algodão here is treated as a material in general, so we use de + [bare noun]:

de algodão – (made) of cotton
de ouro – of gold
de vidro – of glass

We only contract de + o / a / os / as when there is a definite article:

de + o → do
de + a → da
etc.

You’d use that if you were talking about the cotton in a more specific way, e.g. do algodão da Índia (from the cotton of India), but for “made of cotton” the simple de algodão is normal.

Could I say Eu prefiro aquela de algodão instead? What’s the difference?

Yes, you can, but the nuance changes:

Eu prefiro a de algodão. – I prefer the cotton one (between clearly identified options, already known to speaker and listener).
Eu prefiro aquela de algodão. – I prefer that cotton one (pointing to or clearly distinguishing one from others, more deictic – “that one over there / that specific one”).

Aquela includes a sense of distance (physically, or just “that one, not this one”), while a is more neutral and just refers back to the earlier mentioned item.

Can I drop Eu and just say Prefiro a de algodão?

Yes, very naturally. In everyday European Portuguese, speakers often omit eu when it’s clear who is speaking:

Prefiro a de algodão. – I prefer the cotton one.

Both forms are correct. Eu prefiro… is slightly more explicit/emphatic; Prefiro… is more neutral and very common.

Can I say only Eu prefiro de algodão without a?

Not in this context. Eu prefiro de algodão by itself is incomplete and sounds wrong.

You need either:

a noun: Eu prefiro camisas de algodão. – I prefer cotton shirts.
or the article standing for the noun: Eu prefiro a de algodão. – I prefer the cotton one.

So de algodão must be attached to something (an explicit noun or an implied one via a/o/as/os).

Is there any difference between European Portuguese and Brazilian Portuguese for this sentence?

Grammatically and lexically, Eu prefiro a de algodão is fine and natural in both European and Brazilian Portuguese.

Main differences:

Pronunciation (see next question)
In fast speech, Brazilians might more often keep the subject Eu than Europeans, but both options exist in both varieties:
- PT-PT: Prefiro a de algodão. (very common)
- PT-BR: Eu prefiro a de algodão. (very common)

But the structure and meaning are the same.

How is Eu prefiro a de algodão pronounced in European Portuguese?

Approximate European Portuguese pronunciation (Lisbon accent):

Eu – /ew/ (like English “ehw”, one syllable)
prefiro – /pɾɨˈfiɾu/
- pre- with a reduced vowel /ɾɨ/ (not a full “pre”)
- stress on -fi-: preFIro
a – usually a reduced, very short /ɐ/
de – often reduced to /d/ or /dɨ/, almost clitic to the next word
algodão – roughly /aɫɡuˈdɐ̃w̃/
- dark l in al-
- stress on dão
- ão is a nasalized vowel plus a glide (like “uh-ow” with nasalization)

Spoken quickly, the whole sentence might sound like:

[ew pɾɨˈfiɾu ɐ d(ɨ) aɫɡuˈdɐ̃w̃]

Is there a more “casual” or common way to say this idea in Portuguese?

Yes, another very common verb is gostar mais de:

Eu prefiro a de algodão. – I prefer the cotton one.
Eu gosto mais da de algodão. – I like the cotton one more.

Key points:

gostar needs de: gostar de algo
Since de + a contracts, you get da: gosto mais da de algodão.

Both prefiro and gosto mais de are natural; prefiro is a bit more direct and concise.