Ég heilsa henni alltaf á morgnana.

Because að heilsa takes the dative case for the person being greeted.

• hana = accusative her
• henni = dative her

So Icelandic says:

• Ég heilsa henni = I greet her / I say hello to her

This is one of those verbs where you have to learn the case together with the verb. A useful pattern is:

• að heilsa e-m = to greet someone, to say hello to someone

Here e-m is a dictionary shorthand meaning someone in the dative.

Because Icelandic uses the verb heilsa differently from English.

In English, you often say:

• say hello to her

But in Icelandic, heilsa already includes that idea, so you just put the person after the verb in the required case:

• Ég heilsa henni

So the structure is closer to:

• I greet her

rather than word-for-word I say hello to her.

Heilsa is the present tense, first person singular form of að heilsa.

So:

• að heilsa = to greet
• ég heilsa = I greet
• þú heilsar = you greet
• hann/hún/það heilsar = he/she/it greets

In the sentence Ég heilsa henni alltaf á morgnana, the speaker is talking about a regular habit, so the present tense is the natural choice.

Á morgnana means in the mornings or every morning.

It uses a plural form because Icelandic often uses this kind of expression for things that happen repeatedly at a certain time of day.

So:

• á morgnana = in the mornings
• á kvöldin = in the evenings
• á næturnar = at night / during the nights

It is best to learn á morgnana as a whole time expression, not just as a literal word-for-word phrase.

This is a very important distinction:

• á morgun = tomorrow
• á morgnana = in the mornings / every morning

So compare:

• Ég heilsa henni á morgun = I will greet her tomorrow
• Ég heilsa henni á morgnana = I greet her in the mornings

They look similar, but they mean very different things.

Alltaf adds the meaning always.

Without it:

• Ég heilsa henni á morgnana

still sounds habitual, like something the speaker does in the mornings.

With alltaf:

• Ég heilsa henni alltaf á morgnana

the meaning is stronger: I always greet her in the mornings.

So alltaf emphasizes that this happens regularly and consistently, not just generally.

Yes, the word order can change, but the given order is very natural.

The sentence is:

• Ég heilsa henni alltaf á morgnana

A rough breakdown is:

• Ég = subject
• heilsa = verb
• henni = object
• alltaf = frequency adverb
• á morgnana = time expression

This is a normal, neutral order.

You can move parts of the sentence for emphasis, but Icelandic still follows the verb-second rule in main clauses. For example:

• Á morgnana heilsa ég henni alltaf.

That still means basically the same thing, but now á morgnana is emphasized or presented as the topic.

So the original sentence is not the only possible order, just the most straightforward one.

In normal Icelandic, ég is usually necessary.

Unlike some languages, Icelandic does not normally leave out subject pronouns in ordinary sentences. So:

• Ég heilsa henni alltaf á morgnana = normal
• Heilsa henni alltaf á morgnana = usually incomplete on its own

Even though the verb form gives some information, Icelandic normally keeps the pronoun.

It means a general habit.

The present tense in Icelandic, just like in English, can be used for things that happen regularly:

• Ég heilsa henni alltaf á morgnana = I always greet her in the mornings

It does not mean that the speaker is greeting her at this exact moment. The words alltaf and á morgnana make it clear that this is a repeated action.