Hänen päänsä oli kipeä.

Because Finnish hän is gender-neutral. It refers to he or she, and hänen is its possessive/genitive form, so it can mean:

• his
• her

So Hänen päänsä oli kipeä can mean either:

• His head was sore / hurting
• Her head was sore / hurting

You only know which one from context.

In standard Finnish, third-person possession is usually shown in two places:

1. the possessor word: hänen = his/her
2. the possessive suffix on the noun: -nsä = his/her/their

So:

• hänen päänsä = his/her head

This double marking is normal in standard Finnish.

A rough breakdown:

• hänen = his/her
• pää = head
• -nsä = third-person possessive suffix

So the whole noun phrase literally works like his/her head.

Päänsä is the noun pää (head) with a third-person possessive ending.

So:

• pää = head
• päänsä = his/her head, their head, or his/her own head, depending on context

In this sentence, because we already have hänen, the meaning is clearly his/her head.

Also, the form looks a bit different from plain pää because Finnish adds the possessive ending to the noun stem, producing päänsä.

Because oli is the past tense of olla (to be).

• on = is
• oli = was

So:

• Hänen päänsä on kipeä = His/her head is sore
• Hänen päänsä oli kipeä = His/her head was sore

Here kipeä means something like:

• sore
• painful
• hurting

With a body part, kipeä usually means that the body part hurts.

So Hänen päänsä oli kipeä is naturally understood as:

• His/her head was hurting
• His/her head was sore

It does not mean that the head was sick in a strange literal English sense.

Because it is a predicate adjective after olla (to be).

In Finnish, when you say X is Y, the adjective often appears in the nominative basic form:

• Talo on suuri. = The house is big.
• Pää oli kipeä. = The head was sore.

So kipeä is in its normal dictionary form because it is describing the subject through the verb oli.

The subject is päänsä.

So the structure is:

• Hänen = possessor (his/her)
• päänsä = subject noun phrase (head with possessive marking)
• oli = verb (was)
• kipeä = predicate adjective (sore/hurting)

In other words, the sentence is built around his/her head being the thing described.

Sometimes Finnish can leave out the possessor if the context already makes it clear, but in a standalone sentence Päänsä oli kipeä would usually sound odd or incomplete.

Hänen päänsä makes the possessor explicit, so it is clear whose head we are talking about.

So for a clear standard sentence, Hänen päänsä oli kipeä is a good form.

The neutral word order is:

• Hänen päänsä oli kipeä.

That is the most straightforward way to present the information.

Finnish word order is fairly flexible, but changing it changes the focus or emphasis. For example, another order might sound marked, contrastive, or literary.

For learners, this sentence is best understood as a normal neutral pattern:

possessor + noun + verb + adjective

Because Finnish has no articles.

English needs words like:

• a
• an
• the

Finnish does not. So pää can mean:

• a head
• the head

and the exact meaning comes from context.

In Hänen päänsä oli kipeä, English naturally uses his/her head, without needing a separate article in Finnish.

It is a natural sentence, but its meaning is slightly broader than just had a headache.

Hänen päänsä oli kipeä focuses on the body part:

• His/her head was sore
• His/her head was hurting

If you specifically want to say He/she had a headache, Finnish often uses expressions like:

• Hänellä oli päänsärkyä.
• Häntä särki päähän.

So your sentence is correct and natural, but it emphasizes the head hurting, not necessarily the medical label headache.

It can do both.

The suffix -nsa / -nsä is used for third person possession, and by itself it can mean:

• his
• her
• their

So:

• hänen päänsä = his/her head
• heidän päänsä = their head

That is why the separate word before it matters:

• hänen tells you the possessor is singular
• heidän tells you the possessor is plural

In your sentence, hänen makes it clearly his/her head.