Одного аргумента было мало, чтобы убедить брата не спорить с начальницей.

Because мало requires the genitive case: мало чего? = little/not enough of what?

So the phrase is built as:

• одного — genitive singular of один
• аргумента — genitive singular of аргумент

Literally, it is something like of one argument, it was too little.

This is very common with words like:

• мало — not enough / little
• много — much / many
• достаточно — enough

So:

• Одного аргумента было мало = One argument was not enough

Because this is an impersonal construction.

In Russian, words like мало, много, достаточно, жаль, нужно often go with a default past-tense form in neuter singular:

• было мало
• было достаточно
• было нужно

There is no normal nominative subject here controlling agreement. So Russian uses the default impersonal form было.

Here мало is not an adjective. It is a predicative word/adverb-like word meaning too little / not enough.

So:

• Одного аргумента было мало
  = One argument was too little / was not enough

It describes the whole situation, not the noun directly.

Compare:

• мало времени — little time
• мало денег — little money
• этого мало — this is not enough

Брата is accusative singular because it is the direct object of убедить:

• убедить кого? — брата

For animate masculine nouns, the accusative form looks the same as the genitive:

• nominative: брат
• genitive: брата
• accusative: брата

So here it is accusative, even though it looks like genitive.

Убедить is perfective, and that fits the idea of a result: successfully persuading someone.

The sentence says that one argument was not enough to persuade the brother. That points to an intended or possible completed result, so убедить is natural.

Compare:

• убеждать — to be persuading, to try to persuade, to persuade repeatedly
• убедить — to persuade successfully, to convince

After чтобы, perfective infinitives are very common when the idea is achieving a result.

Because спорить is the normal verb for to argue / to be arguing, and after убедить Russian often uses an infinitive for the action someone is being persuaded to do or not do:

• убедить кого-то сделать что-то
• убедить кого-то не делать что-то

So:

• убедить брата не спорить = to persuade the brother not to argue

The imperfective спорить works well because it refers to the activity in general.
Не поспорить would sound wrong here, because поспорить means something more like to have an argument / argue for a while, not the general action being discouraged.

Because the verb спорить takes с + instrumental when you say who someone is arguing with:

• спорить с кем? — с начальницей

So:

• с — with
• начальницей — instrumental singular of начальница

Examples:

• спорить с другом — argue with a friend
• спорить с учителем — argue with a teacher
• спорить с начальницей — argue with the boss

In this sentence, начальница clearly marks the boss as female.

• nominative: начальница
• instrumental: начальницей

Yes, in real Russian, some speakers do use начальник even for a woman, especially in more official or traditional contexts. But начальница is perfectly normal when you want to show the boss is female.

So here с начальницей simply means with his female boss.

Чтобы introduces a clause of purpose or intended result:

• ..., чтобы убедить брата...
• ..., in order to persuade the brother...

So the structure is:

• Одного аргумента было мало — One argument was not enough
• чтобы убедить брата... — to persuade the brother...

This is a very common Russian pattern.

Because чтобы introduces a subordinate clause, and Russian normally separates subordinate clauses with a comma.

So:

• Одного аргумента было мало, чтобы...

This is standard Russian punctuation.

Russian word order is flexible, and the beginning of the sentence often shows what is being highlighted.

Starting with Одного аргумента emphasizes the thing that turned out to be insufficient:

• Одного аргумента было мало...
  = As for one argument, that wasn’t enough...

It sounds natural and idiomatic. Russian often puts the most relevant or contrastive information first.

Yes. It is a very common pattern:

• убедить кого-то сделать что-то
• убедить кого-то не делать что-то

Examples:

• убедить друга прийти — persuade a friend to come
• убедить сына учиться — persuade one’s son to study
• убедить брата не спорить — persuade one’s brother not to argue

So this sentence uses a very standard Russian construction.