Meine Schwester möchte lieber einen unbefristeten Mietvertrag, auch wenn die Wohnung etwas teurer ist.

Together, möchte lieber often means would prefer.

• möchte = would like
• lieber = rather / preferably

So Meine Schwester möchte lieber einen unbefristeten Mietvertrag is very naturally understood as My sister would prefer an open-ended lease.

Because Meine Schwester is the subject of the sentence, so it is in the nominative case.

• meine Schwester = nominative
• meiner Schwester = dative or genitive, depending on context

Here, your sister is the one doing the wanting, so nominative is required.

Because this is the direct object of möchte. In other words, it is the thing your sister wants.

Mietvertrag is a masculine noun, so in the accusative singular you get:

• ein Mietvertrag = nominative
• einen Mietvertrag = accusative

That is why the article is einen.

Because it is an adjective describing a masculine accusative singular noun after einen.

The noun phrase is:

• einen = masculine accusative singular article
• unbefristeten = adjective with the correct ending
• Mietvertrag = noun

After einen, an adjective before a masculine accusative singular noun normally takes -en.

So:

• ein unbefristeter Mietvertrag = nominative
• einen unbefristeten Mietvertrag = accusative

Unbefristet means not limited to a fixed period or open-ended.

For a rental contract, it means the lease does not automatically end on a specific date.

It is the opposite of:

• befristet = fixed-term / temporary

So:

• ein befristeter Mietvertrag = a fixed-term lease
• ein unbefristeter Mietvertrag = an open-ended lease

Mietvertrag is a compound noun:

• Miete = rent
• Vertrag = contract

So literally it is a rental contract, often translated as lease in English.

German uses compound nouns very often, so this is a useful pattern to notice.

Because lieber is an adverb, not an adjective here. It modifies the idea of preference, not the noun itself.

In German main clauses, adverbs like this often appear in the middle of the sentence, after the conjugated verb and before the object:

• Meine Schwester möchte lieber einen unbefristeten Mietvertrag.

That word order sounds natural and standard.

Auch wenn means even if, even though, or sometimes although, depending on context.

In this sentence, it introduces a contrast:

• she would prefer an open-ended lease
• even though the apartment is a bit more expensive

So it shows that the second idea does not change the first one.

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

Main clause:

• Meine Schwester möchte lieber einen unbefristeten Mietvertrag

Subordinate clause:

• auch wenn die Wohnung etwas teurer ist

That final ist is completely normal German subordinate-clause word order.

Because auch wenn starts a subordinate clause, and German normally separates subordinate clauses with a comma.

So the comma marks the boundary between:

• the main clause
• the subordinate clause

This is required in standard written German.

Because die Wohnung is the subject of the subordinate clause.

In die Wohnung etwas teurer ist, the apartment is the thing that is more expensive.

So:

• die Wohnung = subject, nominative
• ist = verb
• etwas teurer = predicate adjective phrase

Etwas teurer means a bit more expensive or somewhat more expensive.

• etwas here means somewhat / a little / a bit
• teurer is the comparative form of teuer = expensive

So:

• teuer = expensive
• teurer = more expensive

Because teurer is not being used directly before a noun. It is used predicatively, after ist.

Compare:

• eine teurere Wohnung = an adjective before a noun, so it needs an ending
• die Wohnung ist teurer = predicate adjective after ist, so no ending

In this sentence, it is the second type:

• die Wohnung ist etwas teurer

So teurer stays without an ending.