Min datter vil have en kanin, men min søn vil hellere have en hamster.

Yes. Min means my.

In Danish, possessive words like my change according to the gender and number of the noun:

• min = for common-gender singular nouns
• mit = for neuter singular nouns
• mine = for plural nouns

So here:

• min datter = my daughter
• min søn = my son

More examples:

• min bil = my car
• mit hus = my house
• mine børn = my children

Also, when you use min, you do not use en or et as well. So min datter is correct, not en min datter.

Because en is the indefinite article for common-gender nouns, like English a/an.

So:

• en kanin = a rabbit
• en hamster = a hamster

In Danish, singular nouns are usually either:

• en-words (common gender)
• et-words (neuter)

Both kanin and hamster are en-words.

Compare:

• en kanin = a rabbit
• et hus = a house

If you wanted the definite form, Danish usually adds the article to the end of the noun:

• kaninen = the rabbit
• hamsteren = the hamster

This is a very common question. On its own, vil often corresponds to English will or wants to, depending on context.

In the expression vil have, Danish often means:

• want
• want to have

So:

• Min datter vil have en kanin = My daughter wants a rabbit

Word-for-word, it looks like my daughter will have a rabbit, but in normal Danish this often expresses desire, not just future time.

A few examples:

• Jeg vil have kaffe. = I want coffee.
• Han vil have en ny cykel. = He wants a new bike.

So here, you should understand vil have as wants / wants to have.

Because after a modal verb in Danish, the next verb usually appears in the infinitive without at.

Vil is a modal verb, so you say:

• vil have
• vil gå
• kan komme
• skal arbejde

Not:

• vil at have ❌

This is similar to English, where we say:

• will go
• can come

and not will to go.

So in the sentence:

• vil have = want/will have

the verb have is an infinitive following the modal vil.

Hellere means rather.

So:

• min søn vil hellere have en hamster = my son would rather have a hamster

It shows a preference, usually between alternatives.

In this sentence, the implied contrast is:

• the daughter wants a rabbit,
• but the son would rather have a hamster.

A useful comparison:

• gerne = gladly
• hellere = rather
• helst = preferably / most of all

Examples:

• Jeg vil gerne have te. = I would like tea.
• Jeg vil hellere have kaffe. = I would rather have coffee.
• Jeg vil helst have vand. = I would prefer water most of all.

Because in a normal Danish main clause, the finite verb usually comes in second position. This is often called the V2 rule.

In the second clause:

• min søn = subject
• vil = finite verb
• hellere = adverb
• have = infinitive

So the order is:

• Min søn vil hellere have en hamster.

This is the normal main-clause pattern: subject + finite verb + adverb + infinitive/object

Compare:

• Jeg vil gerne spise nu.
• Hun kan godt lide musik.
• Vi skal måske gå.

So vil stays early in the clause, and hellere comes after it.

Because men means but, and here it joins two main clauses:

• Min datter vil have en kanin
• men min søn vil hellere have en hamster

In standard Danish writing, a comma before men is normal here.

So the comma helps show the contrast between the two complete statements.

You can think of it like English:

• My daughter wants a rabbit, but my son would rather have a hamster.

In this sentence, yes: men means but.

It is a coordinating conjunction used to connect two clauses or ideas in contrast.

So:

• A, men B = A, but B

Examples:

• Jeg er træt, men glad. = I am tired, but happy.
• Han vil gå, men hun vil blive. = He wants to leave, but she wants to stay.

So in your sentence, men introduces the contrast between what the daughter wants and what the son prefers.

The letter ø is one of the special Danish vowels, and English does not have an exact equivalent.

In søn, the ø sounds somewhat like:

• the vowel in German schön
• or French bleu

But not exactly like English o in son or go.

A rough guide:

• søn sounds approximately like surn or sirn said with rounded lips, but that is only an approximation.

A few pronunciation notes:

• søn = one syllable
• datter has a soft final -er, something like a weak uh
• kanin is stressed on the last syllable: ka-NIN
• hamster usually has the stress on the first syllable: HAM-ster

If you are learning pronunciation, ø is definitely worth practicing separately.

Here are the nouns from the sentence:

• en datter = a daughter
• en søn = a son
• en kanin = a rabbit
• en hamster = a hamster

Their definite singular forms are:

• datteren = the daughter
• sønnen = the son
• kaninen = the rabbit
• hamsteren = the hamster

This is a very common Danish pattern: instead of putting a separate word for the in front, Danish often adds the definite ending to the noun itself.

So:

• en kanin / kaninen
• en søn / sønnen

That is an important feature of Danish noun grammar.