Learn Hardcore Icelandic: Hún býður okkur í mat á morgun.

bjóða + DAT (person) + í + ACC (event/place) = invite someone to something. Example pattern: bjóða okkur í mat (invite us for a meal).
bjóða + DAT (person) + ACC (thing) = offer someone something. Example: Hún býður okkur kaffi (She offers us coffee).
bjóða upp á + ACC = offer/serve, make available. Example: Þeir bjóða upp á súpu (They offer/serve soup).

Why is okkur used, and what case is it?

Okkur is the dative form of “we/us.” The verb bjóða takes the person being invited in the dative. Other common dative pronouns:

mér (me), þér (you sg.), honum/henni/því (him/her/it),
okkur (us), ykkur (you pl.), þeim (them).

What does í mat mean, literally and idiomatically, and why í?

Literally “into/for meal,” idiomatically “for a meal” (i.e., for dinner/lunch, context decides). The preposition í often takes the accusative to show movement/goal, and this is a fixed invitation phrase: bjóða e-m í mat. The noun matur (meal/food) is in the form mat here (its accusative).

Is the present tense normal for a future meaning with á morgun?

Yes. Icelandic often uses the present tense for near-future or scheduled events when there’s a time adverbial like á morgun (tomorrow). Alternatives:

Hún ætlar að bjóða… (she intends/plans to invite…)
Hún mun bjóða… (she will invite…) — more predictive/formal. All are possible; the simple present is very natural.

Can the time phrase move, and what about verb position?

Yes. Two common orders:

Hún býður okkur í mat á morgun. (time at the end)
Á morgun býður hún okkur í mat. (time first) Icelandic main clauses are V2: the finite verb (býður) stays in second position in the clause.

How do I make this a yes–no question?

Invert subject and verb:

Býður hún okkur í mat á morgun?

How do I negate it?

Place ekki after the finite verb (and after unemphatic pronouns):

Hún býður okkur ekki í mat á morgun.

Why á morgun and not í morgun?

á morgun = tomorrow.
í morgun = this morning (earlier today). They are not interchangeable.

Does á morgun literally mean “on the morning”? Is that the same as “in the mornings”?

Historically it looks like “on (the) morning,” but as a fixed adverbial it simply means “tomorrow.” For habitual time you’d use:

á morgnana = in the mornings (regularly).

How can I specify dinner vs. lunch vs. coffee?

Use a more specific noun:

í kvöldmat = for dinner,
í hádegismat = for lunch,
í morgunmat = for breakfast,
í kaffi = for coffee (a coffee visit).

What’s the difference between í mat and í matinn?

í mat = for a meal (neutral/default).
í matinn = to the (specific) meal/dinner, used when a particular meal is understood from context. In invitations, the indefinite í mat is most common.

What’s the difference between bjóða okkur í mat and bjóða okkur mat?

bjóða okkur í mat = invite us for a meal (to come over).
bjóða okkur mat = offer us food (hand/give/serve food). Different meaning and construction.

Can I put okkur later, like Hún býður í mat okkur?

No; that word order is unnatural. Keep the dative person right after the verb:

Natural: Hún býður okkur í mat…
Unnatural: Hún býður í mat okkur…

What are some useful other forms of bjóða?

Present: ég býð, þú býður, hún/hann býður, við bjóðum, þið bjóðið, þau/þeir/þær bjóða.
Past: ég/hún bauð, við buðum, þau buðu.
Past participle: boðið (as in mér var boðið = I was invited/offered).

Any quick pronunciation tips for the sentence?

Approximate guide:

Hún ≈ “hoon” (long u).
býður ≈ “BEE-thur” (ý = long ee; ð = voiced “th” as in “this”).
okkur ≈ “OHK-kur” (kk sounds like hk).
í = “ee.”
mat ≈ “maht.”
á ≈ “ow” in “cow.”
morgun ≈ “MOR-guhn” (the g after r is often a softer sound; u is short, like German ü without rounding). Don’t stress too much over fine details; rhythm and vowel length matter most.

Is býður the same as biður?

No.

býður is from bjóða (invite/offer).
biður is from biðja (ask/request). They look similar but are different verbs with different meanings.