Hún finnur notendanafnið sitt í bréfi í pósthólfinu sínu.

Because hún is the subject of the sentence (the person doing the finding).

• hún = nominative (subject case)
• hana = accusative (direct object)
• henni = dative (indirect object / after many prepositions)

Here, hún finnur ... = she finds ..., so nominative is required.

finnur is the 3rd person singular present tense form of að finna (to find).
Conjugation (present) is typically:

• ég finni
• þú finnur
• hún/hann/það finnur
• við finnum
• þið finnið
• þeir/þær/þau finna

So hún finnur matches she finds.

Icelandic often forms compound nouns. notendanafnið is:

• notenda- = user (from notandi, here in a linking/plural-genitive-like form)
• nafn = name
• -ið = the definite article (the)

So it’s essentially the username packed into one word.

That -ið is the suffixed definite article for a neuter noun in the singular (nominative/accusative).

• nafn = a name
• nafnið = the name
  Likewise:
• notendanafn = a username
• notendanafnið = the username

Icelandic usually attaches the to the end of the noun instead of using a separate word.

In Icelandic, possessives commonly come after the noun, especially in neutral, straightforward prose.
So notendanafnið sitt = her (own) username, with the possessive following what it modifies.

You can sometimes see other placements in poetry or for emphasis, but this post-noun position is very normal.

Because sitt / sínu are reflexive possessives, used when the possessor is the subject of the clause. The subject is hún, so her possessions take reflexive forms:

• notendanafnið sitt = her own username (belonging back to the subject)
• pósthólfinu sínu = her own mailbox (again referring back to the subject)

hennar is a non-reflexive possessive and tends to mean her in the sense of someone else’s (or used when the possessor is not the subject of that clause). Reflexive vs non-reflexive can change meaning in Icelandic more sharply than in English.

They agree with the noun they modify in case, number, and gender.

• notendanafnið is neuter singular and here it’s in the accusative (direct object of finnur), so you get sitt.
• pósthólfinu is neuter singular dative (because of the preposition í here), so you get sínu (dative form).

Same idea, different grammatical context.

With í, Icelandic chooses case based on meaning:

• í + accusative often implies movement into something (into)
• í + dative often implies location/state (in)

Here it means the username is located in the letter, not moving into it, so it’s í bréfi (dative of bréf).

Same rule as above: it’s describing where the letter is (a location), so í takes dative:

• pósthólf (neuter) → dative singular definite: pósthólfinu

So í pósthólfinu = in the mailbox (location).

pósthólfinu = pósthólf (mailbox / post box) in dative singular definite.

Pieces:

• base noun: pósthólf (neuter)
• dative singular ending: -i (often seen as pósthólfi when indefinite)
• definite article in dative singular neuter: -nu
  So: pósthólf → pósthólfi (in a mailbox) → pósthólfinu (in the mailbox)

They’re two separate prepositional phrases with different “containers”: 1) í bréfi = in a letter (the username is inside the letter)
2) í pósthólfinu sínu = in her mailbox (the letter is located in the mailbox)

Icelandic often nests locations like this without needing extra linking words.

It attaches specifically to pósthólfinu (the immediately preceding noun phrase):

• pósthólfinu sínu = her (own) mailbox

It doesn’t refer to the letter or the username; it agrees with and modifies pósthólfinu in dative neuter singular.

This order is very natural, but Icelandic has some flexibility because case endings show roles. Still, certain patterns are preferred:

• Subject – verb – object is common in simple main clauses: Hún finnur notendanafnið...
• Prepositional phrases often come after the object, as here.

You could move phrases for emphasis or style, but you’d typically keep:

• Hún finnur together
• the object notendanafnið sitt as a unit
• each í ... phrase intact with correct case endings