Henni var boðið að prófa annað hljómborð áður en tónleikarnir byrjuðu.

Because að bjóða takes the person receiving the invitation or offer in the dative:
að bjóða einhverjum að gera eitthvað = to invite/offer someone to do something.

So the sentence uses henni (dative of hún) rather than hún (nominative). Even in an active sentence, it would still be dative:

Einhver bauð henni að prófa annað hljómborð.
= Someone invited/offered her to try another keyboard.

Var boðið is a passive/impersonal passive-style construction of að bjóða in the past.

• var = was
• boðið = past participle form of bjóða

So Henni var boðið... means something like She was invited/offered... or more literally To her was offered...

This kind of structure is very common in Icelandic when the person doing the action is not mentioned.

Because this sentence does not have a normal nominative subject for the participle to agree with. In that situation, Icelandic often uses the default neuter singular form of the participle.

So even though the sentence is about her, the participle does not become feminine. It stays boðið.

This is one reason Icelandic passives can feel a bit different from English ones.

It can be understood as either, depending on context.

With að + infinitive, bjóða often means:

• invite someone to do something
• offer someone the chance to do something

So Henni var boðið að prófa... could be understood as:

• She was invited to try...
• She was offered the chance to try...

Both are reasonable. The surrounding context would tell you which nuance is stronger.

Because að is the normal infinitive marker, like English to.

So:

• að prófa = to try

After bjóða, Icelandic can use an infinitive clause:

bjóða einhverjum að gera eitthvað
= invite/offer someone to do something

That is exactly what is happening here.

Because hljómborð is a neuter noun, and annar must agree with it.

So:

• masculine: annar
• feminine: önnur
• neuter: annað

Here annað hljómborð means another keyboard or a different keyboard.

Also, prófa takes a direct object, so hljómborð is functioning as the object of prófa. In the neuter singular, nominative and accusative look the same, so the form does not change.

Not necessarily. Here it most naturally means another or a different one.

The word annar can sometimes relate to the idea of other/another/second, but in this sentence the most natural reading is simply that she was going to try a different keyboard from the one already being used or considered.

Because tónleikar is a plural-only noun in Icelandic. Even when English says the concert, Icelandic normally uses the plural form:

• tónleikar = concert
• tónleikarnir = the concert

Since the noun is grammatically plural, the verb must also be plural:

• tónleikarnir byrjuðu = the concert began

So the plural verb is required by Icelandic grammar.

Áður en means before and introduces a subordinate clause.

So:

• áður en tónleikarnir byrjuðu
  = before the concert began

After áður en, you get a full clause with its own subject and verb. Here:

• subject: tónleikarnir
• verb: byrjuðu

Because the sentence is describing a real past event: the concert actually began.

So Icelandic uses the normal past indicative:

• byrjuðu = began / started

In more hypothetical or uncertain contexts, other moods can appear in subordinate clauses, but here the meaning is straightforward and factual.

Because Icelandic often puts the most important or topical element first. Here the sentence is centered on her, so Henni is placed at the front.

This is also natural because the person doing the inviting/offering is not mentioned. Starting with Henni makes the sentence feel focused on the person affected:

Henni var boðið...
= She was invited/offered...

The finite verb still stays in second position, which is normal Icelandic word order:

• Henni = first element
• var = second element

A natural active version would be:

Einhver bauð henni að prófa annað hljómborð áður en tónleikarnir byrjuðu.

That means:

Someone invited/offered her to try another keyboard before the concert began.

This shows the underlying roles clearly:

• einhver = the person making the offer/invitation
• henni = the recipient, still in dative
• að prófa annað hljómborð = what she was invited/offered to do