Min søn vil hellere blive på legepladsen lidt længere.

Not mainly. In this sentence, vil is part of vil hellere, which means would rather or would prefer to.

So Min søn vil hellere... is not really about the future in the English will sense. It expresses your son’s preference.

A useful way to think about it:

• vil = wants to / would like to / will, depending on context
• vil hellere = would rather

Hellere means rather.

It is the comparative form related to gerne:

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

So:

• Jeg vil gerne blive = I would like to stay
• Jeg vil hellere blive = I would rather stay
• Jeg vil helst blive = I would prefer most of all to stay

In your sentence, hellere shows preference.

Yes, blive can mean become, but it also very often means stay or remain.

In this sentence, because it is followed by a place expression, på legepladsen, it means stay:

• blive hjemme = stay home
• blive i sengen = stay in bed
• blive på legepladsen = stay at/on the playground

So the meaning is not become on the playground. It is stay on the playground.

Because vil is a modal verb, and after modal verbs Danish normally uses the bare infinitive without at.

Common modal verbs include:

• kan = can
• skal = must / shall
• må = may / be allowed to
• vil = want to / will
• bør = ought to

So:

• Han vil blive = He wants to stay / He will stay
• not Han vil at blive

This is similar to English:

• He can go
• not He can to go

Because after a possessive like min, the noun normally stays in the indefinite form.

So:

• min søn = my son
• din bil = your car
• vores hus = our house

You do not usually combine min with the suffixed definite ending:

• not min sønnen

That is because min already makes the noun definite in meaning, just as my son does in English.

Because legepladsen is the definite singular form: the playground.

The base noun is:

• en legeplads = a playground

The definite form is:

• legepladsen = the playground

Danish usually adds definiteness to the end of the noun instead of using a separate word like the.

So:

• en legeplads = a playground
• legepladsen = the playground

Because på is the natural preposition with legeplads in this meaning.

Danish often uses på with places and areas where an activity happens, and legeplads is one of those common cases:

• på legepladsen = at/on the playground

Using i would usually sound less natural here, unless you are emphasizing being physically inside a bounded area in a very literal way.

English actually does something similar:

• on the playground

So på legepladsen is the normal idiomatic choice.

It means a little longer.

• lidt = a little
• længere = longer / for longer

Together:

• lidt længere = a bit longer / a little longer

In this sentence it refers to duration:

• blive ... lidt længere = stay a little longer

Here it is functioning as an adverb, because it describes how long the staying lasts.

It is not describing a noun. It is describing the verb blive.

Compare:

• en længere pause = a longer break
  Here længere is adjective-like, because it describes pause.
• blive længere = stay longer
  Here længere is adverbial, because it describes the action.

In this sentence, længere means for a longer time.

Because Danish word order usually places the finite verb early in the clause and puts adverbs like hellere before the main infinitive.

So the structure is:

• Min søn = subject
• vil = finite verb
• hellere = adverb
• blive = infinitive
• på legepladsen = place
• lidt længere = duration

This is a very normal Danish pattern:

• Han vil gerne spise nu
• Hun kan ikke komme
• Vi vil hellere vente

You can partly do that, but a fully natural English translation usually needs a small adjustment.

Word by word, it looks something like:

• Min søn = my son
• vil hellere = would rather
• blive = stay
• på legepladsen = on the playground
• lidt længere = a little longer

A natural English version would be:

• My son would rather stay on the playground a little longer.

So the Danish is quite close to English here, but vil hellere works as one idea, not as separate will rather in normal English.