Η κόρη της θέλει κακάο το πρωί, αλλά ο γιος της προτιμά γάλα.

They are the Greek definite articles, meaning the.

• η goes with κόρη because κόρη is feminine singular.
• ο goes with γιος because γιος is masculine singular.

So:

• η κόρη = the daughter
• ο γιος = the son

Greek articles change for gender, number, and case.

In Greek, a very common way to show possession is to put a weak genitive pronoun after the noun.

So:

• η κόρη της = her daughter
• ο γιος της = her son

This is normal Greek word order for this kind of possession. English says her daughter, but Greek usually says something more like the daughter of hers.

Because της refers to the possessor, not to the thing possessed.

Here, της means her. It stays the same whether the possessed noun is:

• feminine: η κόρη της
• masculine: ο γιος της
• even neuter: το παιδί της

So the gender of κόρη or γιος does not change της.

Here is the basic breakdown:

• η κόρη — nominative, because it is the subject
• της — genitive, showing possession
• κακάο — accusative, because it is the direct object of θέλει
• το πρωί — accusative in a time expression
• ο γιος — nominative, because it is the subject
• της — genitive again, showing possession
• γάλα — accusative, because it is the direct object of προτιμά

A useful thing to notice: κακάο and γάλα are neuter nouns, and neuter nominative and accusative often look the same, so you do not see a form change there.

Because Greek often leaves out the article with foods and drinks when speaking in a general sense.

So:

• θέλει κακάο = wants cocoa
• προτιμά γάλα = prefers milk

This sounds natural when talking about what someone wants or prefers in general. If you added an article, it could sound more specific, depending on context.

Because Greek often uses the definite article in time expressions.

• το πρωί = in the morning
• το βράδυ = in the evening / at night
• το απόγευμα = in the afternoon

So even though English often says in the morning, Greek uses the morning as a fixed time expression.

Both are present tense, third person singular forms.

• θέλει comes from θέλω = to want
• προτιμά comes from προτιμώ or προτιμάω = to prefer

They are third person singular because the subjects are singular:

• η κόρη της → θέλει
• ο γιος της → προτιμά

So the verbs agree with their subjects.

Because Greek can often leave that idea understood from context.

If someone says:

• θέλει κακάο
• προτιμά γάλα

it is natural to understand wants cocoa to drink or prefers milk. The action is obvious, so Greek does not need to spell out drink.

English can do something similar too: She wants coffee does not need to drink either.

Not completely. Greek word order is more flexible than English word order.

The sentence as given is neutral and natural:

• Η κόρη της θέλει κακάο το πρωί, αλλά ο γιος της προτιμά γάλα.

But Greek could move parts around for emphasis, for example:

• Το πρωί, η κόρη της θέλει κακάο...
• Ο γιος της, όμως, προτιμά γάλα.

Even so, some words usually stay close together, especially noun phrases like:

• η κόρη της
• ο γιος της

Αλλά means but and introduces a contrast.

The sentence is contrasting two preferences:

• the daughter wants cocoa in the morning
• but the son prefers milk

So αλλά works very much like English but.

Because in Greek, a comma is normally used before αλλά when it joins two contrasting clauses.

So the punctuation here is standard:

• ..., αλλά ...

This is similar to English, where a comma often appears before but when it joins two full clauses.

Yes, but that adds emphasis.

• η κόρη της = normal, neutral her daughter
• η δική της κόρη = her own daughter / her daughter specifically

Likewise:

• ο γιος της = neutral
• ο δικός της γιος = more emphatic

In your sentence, the simple forms η κόρη της and ο γιος της are the most natural choice.