Ég lofa að hringja í þig á morgun.

Icelandic normally keeps the subject pronoun, so Ég lofa... is the standard, neutral way to say it. You can drop ég in very informal speech (especially if it’s obvious from context), but it’s much less common than in languages like Spanish or Italian.

Lofa is the present tense, 1st person singular: ég lofa = I promise.
The present tense of lofa is:

• ég lofa
• þú lofar
• hann/hún/það lofar
• við lofum
• þið lofið
• þeir/þær/þau lofa

Here að introduces an infinitive clause: lofa að + infinitive = promise to + verb.
So Ég lofa að hringja... literally means I promise to call....
This að is very common after many verbs that take an infinitive (similar to English to).

Yes. Hringja is the infinitive (to call / to ring).
So að hringja functions like to call in English, as the complement of lofa.

In Icelandic, the “telephone call” meaning uses the fixed pattern hringja í + accusative:

• hringja í einhvern = call someone (on the phone)

Without í, hringja is more like ring (a bell) or ring in a more literal sense, and it won’t sound like the normal verb pattern for phoning someone.

Þig is the accusative form of þú (you, singular).
It’s accusative because í (in this usage with hringja í) governs the accusative:

• í + þig (acc.)

For reference:

• nominative: þú
• accusative: þig
• dative: þér
• genitive: þín

Þig is singular only. Plural you would be ykkur in this construction:

• Ég lofa að hringja í ykkur á morgun. = I promise to call you (plural) tomorrow.

Á morgun is the normal idiomatic way to say tomorrow in Icelandic.
Historically, á often appears with time expressions (roughly on/at), and á morgun has become a set phrase meaning tomorrow.

It’s accusative singular (morgun, masculine). In many time expressions, á can take the accusative, and á morgun is learned best as a fixed expression.

Main clauses in Icelandic generally follow V2 (verb-second) word order: the finite verb (lofa) comes early in the clause.
So you get:

• Ég (subject) + lofa (finite verb) + the rest (að hringja...)

The infinitive hringja stays later in the infinitive clause after að.

You usually put ekki after the finite verb:

• Ég lofa ekki að hringja í þig á morgun. = I don’t promise to call you tomorrow.

If you mean you promise not to call:

• Ég lofa að hringja ekki í þig á morgun. = I promise not to call you tomorrow.

Stress in Icelandic is typically on the first syllable of words:

• ÉG LO-fa að HRING-ja í þig á MOR-gun

A few useful pronunciation notes:

• Ég: the g is very soft; many speakers make it sound like yeh or yehg.
• hr- in hringja is a voiceless r sound (a distinctive Icelandic feature).
• þ in þig is like th in thin (not this).

Yes. A very common alternative is using skulu for a promise-like intention:

• Ég skal hringja í þig á morgun. = I’ll call you tomorrow (often with a reassuring/promise vibe)

But Ég lofa að... is the most direct match to I promise to....