Mit einer Gutenachtgeschichte kann ich meinen Sohn oft schneller trösten als mit einem Lied.

German often puts something other than the subject in the first position to set the topic or add emphasis. Here, Mit einer Gutenachtgeschichte highlights the method: with a bedtime story.

So:

Mit einer Gutenachtgeschichte kann ich meinen Sohn oft schneller trösten ...

is perfectly normal and means essentially the same as:

Ich kann meinen Sohn mit einer Gutenachtgeschichte oft schneller trösten ...

The difference is mostly one of emphasis and flow.

This is because of the German verb-second rule.

In a main clause, the finite verb must be in the second position. Since Mit einer Gutenachtgeschichte takes the first position, kann must come next, and the subject ich comes after it.

So the structure is:

1. Mit einer Gutenachtgeschichte
2. kann
3. ich
4. the rest of the sentence

Because kann is a modal verb. In German, when you use a modal verb such as können, müssen, wollen, etc., the main verb usually goes to the end in the infinitive.

So:

• kann = finite modal verb
• trösten = infinitive main verb at the end

This is the same pattern as:

• Ich kann kommen.
• Sie will lernen.
• Wir müssen gehen.

Because mit always takes the dative case.

So:

• eine Gutenachtgeschichte becomes mit einer Gutenachtgeschichte
• ein Lied becomes mit einem Lied

The forms are different because the nouns have different genders:

• die Gutenachtgeschichte → feminine → einer
• das Lied → neuter → einem

Because trösten takes a direct object, and that object is in the accusative case.

Here, Sohn is masculine, so the possessive determiner changes:

• nominative: mein Sohn
• accusative: meinen Sohn

Compare:

• Mein Sohn schläft.
  Here mein Sohn is the subject.

• Ich tröste meinen Sohn.
  Here meinen Sohn is the object.

German very often combines nouns into one long compound noun, and all nouns are capitalized.

So Gutenachtgeschichte is one noun meaning bedtime story.

It is built from the idea of Gute Nacht plus Geschichte. In standard writing, you may also see the hyphenated form Gute-Nacht-Geschichte. Both represent a noun, so they are capitalized.

oft means often.

It tells you that this is something that happens frequently: the speaker can often comfort his son faster with a bedtime story than with a song.

Its position is natural in the middle of the sentence, before schneller. German adverbs can move around somewhat, but different positions may change emphasis slightly.

Here, oft sounds very natural where it is.

The sentence compares two ways of comforting the son:

• mit einer Gutenachtgeschichte
• mit einem Lied

So the idea is:

I can often comfort my son faster with a bedtime story than with a song.

German often leaves out repeated words in comparisons. The full expanded version would be something like:

Mit einer Gutenachtgeschichte kann ich meinen Sohn oft schneller trösten als mit einem Lied.

Underlying meaning:

... schneller trösten, als ich es mit einem Lied kann.

But repeating all of that would sound heavy, so German normally omits it.

Because schneller is a comparative form.

In German:

• als is used for inequality / comparison
  schneller als = faster than

• wie is used for equality
  so schnell wie = as fast as

So here:

• schneller als mit einem Lied = faster than with a song

If you used wie, it would be wrong in standard German.

Yes, literally it means with a song.

In context, it usually suggests using a song as a way to comfort the child, most likely by singing it. German often leaves that kind of practical detail unstated because the meaning is clear from context.

So mit einem Lied can be understood as something like:

• by singing a song
• using a song
• with the help of a song

Yes. German word order is flexible, as long as the main rules are respected.

For example, you could also say:

Ich kann meinen Sohn mit einer Gutenachtgeschichte oft schneller trösten als mit einem Lied.

This means the same thing. The original version just puts more emphasis on Mit einer Gutenachtgeschichte by placing it first.

So the choice is mainly about information structure and emphasis, not a change in basic meaning.