Fyrsti stafurinn í nafninu hennar er hástafur.

Because stafur is a masculine noun, and fyrsti is the masculine singular form of fyrstur meaning first.

In this sentence:

• fyrsti = masculine singular
• stafurinn = the letter, masculine singular

If the noun were neuter or feminine, the form would be different. So the ending is matching the gender of stafur.

Stafurinn means the letter.

It is made from:

• stafur = letter
• -inn = the definite article the

So Icelandic often attaches the to the end of the noun instead of using a separate word in front.

Compare:

• stafur = a letter / letter
• stafurinn = the letter

Because the preposition í often takes the dative case when it means in in a static sense.

Here the meaning is in her name, not into her name, so Icelandic uses the dative:

• nafn = name
• nafni = name, dative singular
• nafninu = the name, dative singular definite

So:

• í nafninu hennar = in her name

A very common rule is:

• í
  • dative = in
• í
  • accusative = into

Yes, but this is normal Icelandic structure.

In English, we say her name.
In Icelandic, it is very common to say something more literally like the name her:

• nafnið hennar = her name
• bíllinn minn = my car
• húsið þeirra = their house

So a definite noun plus a following possessive is standard Icelandic. It may look strange from an English point of view, but it is completely natural.

Hennar is the genitive form of hún (she), and it is used to mean her in a possessive sense.

So:

• hún = she
• hana = her
• hennar = her / hers

In this sentence, hennar means her as in her name.

Because Icelandic does not have a separate indefinite article like English a/an.

So:

• hástafur can mean a capital letter
• hástafurinn means the capital letter

After er (is), Icelandic often uses a bare noun when English would use a/an:

• Hann er kennari = He is a teacher
• Þetta er bíll = This is a car
• ... er hástafur = ... is a capital letter

So hástafur is exactly what you would expect here.

Yes. Hástafur is the standard word for capital letter or uppercase letter.

The opposite is:

• lágstafur = lowercase letter

So:

• A is a hástafur
• a is a lágstafur

This is the normal grammatical vocabulary used in Icelandic.

A useful way to break it up is:

• Fyrsti stafurinn = the first letter
• í nafninu hennar = in her name
• er = is
• hástafur = a capital letter

So the whole subject is:

Fyrsti stafurinn í nafninu hennar = The first letter in her name

Then:

er hástafur = is a capital letter

That means í nafninu hennar is describing stafurinn, not hástafur.

Yes. A very natural alternative is:

Nafnið hennar byrjar á hástaf.

That means Her name begins with a capital letter.

The given sentence is still perfectly good Icelandic; it is just a bit more explicit because it focuses on the first letter.

A rough English-friendly pronunciation would be:

FIR-sti STAH-vu-rinn ee NAV-ni-nu HEN-nar er HAU-sta-vur

A few helpful points:

• Stress in Icelandic usually falls on the first syllable
• í is like ee
• á is roughly like ow in now, but longer
• In words like stafur, the f is often pronounced more like v

So stafurinn sounds closer to STAH-vu-rinn than to a hard English f sound.

Because we are talking about one specific letter: the first letter of a particular name.

So Icelandic naturally uses the definite form:

• stafurinn = the letter

This is similar to English, where we would also normally say the first letter, not just first letter, in a full sentence like this one.

The noun is specific and identifiable, so the definite form is the natural choice.