Ich weiß nicht, ob sie lieber Blumenkohl oder Brokkoli essen möchte.

Ob means whether / if in the sense of introducing an indirect yes/no question.

So:

• Ich weiß nicht, ob ... = I don't know whether ...

This is different from English if meaning a condition. In German:

• ob = whether
• wenn = if/when in conditional or repeated-time meanings

So in this sentence, ob is correct because the speaker does not know which option she prefers.

Because ob introduces a subordinate clause, and in German subordinate clauses the conjugated verb usually goes to the end.

Main clause:

• Ich weiß nicht
  Here weiß is in the normal second position.

Subordinate clause:

• ob sie lieber Blumenkohl oder Brokkoli essen möchte
  Here the conjugated/modal-like verb möchte goes to the end.

A useful rule:

• Main clause: verb in 2nd position
• Subordinate clause: conjugated verb at the end

This is because möchte is used with an infinitive.

• möchte = would like
• essen = to eat

So:

• sie ... essen möchte = she would like to eat

In German verb clusters, the infinitive often comes near the end, and the conjugated modal-like verb comes in final position in a subordinate clause.

Compare:

• Main clause: Sie möchte Brokkoli essen.
• Subordinate clause: ..., ob sie Brokkoli essen möchte.

Möchte is the usual polite/common form meaning would like.

It comes from mögen (to like), but in modern German möchte often expresses desire or preference, not just general liking.

So:

• Sie möchte Brokkoli essen = She would like to eat broccoli
• not simply She likes broccoli

In this sentence, möchte works well because the sentence is about which of the two she would prefer to eat.

Lieber means rather or preferably, and it is used to compare preferences.

Here:

• lieber Blumenkohl oder Brokkoli = cauliflower or broccoli rather / preferably

A more natural English interpretation is:

• whether she would prefer cauliflower or broccoli

Related forms:

• gern = gladly / like doing
• lieber = rather / prefer
• am liebsten = most of all

Examples:

• Ich esse gern Gemüse. = I like eating vegetables.
• Ich esse lieber Brokkoli. = I prefer eating broccoli.
• Am liebsten esse ich Kartoffeln. = I most like eating potatoes.

In German, food items are often used without an article when talking about them in a general sense.

So:

• Blumenkohl oder Brokkoli essen = eat cauliflower or broccoli

This is similar to English, where we also often say:

• I eat broccoli not necessarily
• I eat the broccoli

You could use articles in a more specific context:

• den Blumenkohl = the cauliflower
• den Brokkoli = the broccoli

But here the sentence is about the general choice between two foods, so no article sounds natural.

Because they are nouns, and all nouns in German are capitalized.

So:

• Blumenkohl = cauliflower
• Brokkoli = broccoli

This is a basic German spelling rule:

• nouns always begin with a capital letter
• verbs, adjectives, and most other words usually do not

For example:

• ich = not capitalized
• essen = not capitalized
• Brokkoli = capitalized

They are in the accusative case, because they are the direct objects of essen.

The hidden question is:

• What does she want to eat?
• Blumenkohl oder Brokkoli

However, you do not see a special accusative ending here because these nouns do not change form in this context.

If you used an article, you could see the accusative more clearly:

• den Blumenkohl
• den Brokkoli

So the structure is:

• sie = subject
• Blumenkohl oder Brokkoli = object of essen

Grammatically, sie can mean:

• she
• they
• you (formal, but then it would be capitalized: Sie)

In this sentence, sie is lowercase, so it can only mean:

• she or
• they

But the verb möchte is singular, so it matches she, not they.

Compare:

• sie möchte = she would like
• sie möchten = they would like / you would like (formal)

So here sie must mean she.

Both are possible in some contexts, but they mean slightly different things.

• mag = likes
• möchte = would like

Compare:

• Sie mag Brokkoli. = She likes broccoli.
• Sie möchte Brokkoli essen. = She would like to eat broccoli.

In your sentence, the idea is not just her general opinion of these vegetables, but which one she would prefer to eat. That is why möchte fits very well.

You could also say:

• Ich weiß nicht, ob sie Blumenkohl oder Brokkoli lieber mag.

That would mean something more like:

• I don't know whether she likes cauliflower or broccoli better.

So the nuance changes a little.

Yes, German word order is somewhat flexible, but some positions sound more natural than others.

The given sentence:

• ..., ob sie lieber Blumenkohl oder Brokkoli essen möchte.

is natural and clear.

You may also hear:

• ..., ob sie Blumenkohl oder Brokkoli lieber essen möchte.

This is also correct. In both versions, lieber modifies the choice between the two foods.

German often places words like lieber near the thing they are emphasizing, but exact placement can vary.

Because German uses a comma to separate a subordinate clause from the main clause.

So:

• Ich weiß nicht, = main clause
• ob sie lieber Blumenkohl oder Brokkoli essen möchte. = subordinate clause

This comma is required in standard German spelling.

Not in this sentence.

Use ob when the structure is basically whether A or B:

• I don't know whether she prefers A or B.

Use was when you mean what more generally:

• Ich weiß nicht, was sie essen möchte. = I don't know what she wants to eat.

So:

• ob sie lieber Blumenkohl oder Brokkoli essen möchte = whether she would rather eat cauliflower or broccoli
• was sie essen möchte = what she wants to eat

Because the sentence gives two explicit options, ob is the right choice.

No, there are several natural alternatives, depending on the nuance.

For example:

• Ich weiß nicht, ob sie lieber Blumenkohl oder Brokkoli essen möchte.
  I don't know whether she would rather eat cauliflower or broccoli.

• Ich weiß nicht, ob sie lieber Blumenkohl oder Brokkoli mag.
  I don't know whether she likes cauliflower or broccoli better.

• Ich weiß nicht, was sie lieber isst: Blumenkohl oder Brokkoli.
  I don't know which she prefers to eat: cauliflower or broccoli.

Your original sentence is completely natural, especially if the situation is about choosing what to eat now.