Ég er enn óviss um hvort ég eigi að samþykkja boðið.

Because this clause expresses uncertainty, and Icelandic often uses the subjunctive in that situation.

• á = present indicative of eiga
• eigi = present subjunctive of eiga

In hvort ég eigi að samþykkja boðið, the speaker is unsure about what they should do, so the subjunctive fits well.

A rough comparison:

• ég á að samþykkja = I am supposed to accept / I should accept
• hvort ég eigi að samþykkja = whether I should accept

So eigi is not random; it is the expected subjunctive form here.

The adjective óviss normally goes with um when you say what someone is uncertain about.

• vera óviss um eitthvað = to be uncertain about something

That pattern stays the same even when what follows is a whole clause:

• óviss um boðið = uncertain about the offer
• óviss um hvort ég eigi að samþykkja boðið = uncertain about whether I should accept the offer

So:

• hvort = whether
• um is there because óviss regularly takes it

English would usually just say uncertain whether..., but Icelandic keeps the preposition.

Here hvort means whether.

It introduces an indirect yes/no question:

• Hvort á ég að fara? = whether I should go
• Hvort ég eigi að samþykkja boðið = whether I should accept the offer

English often uses if and whether in similar situations, but Icelandic hvort is specifically the normal word for this kind of whether clause.

So in this sentence, think of hvort as:

• whether, not a condition like if in if it rains

No. In this sentence, eiga does not mean to own or to possess.

Here it is part of the very common construction:

• eiga að + infinitive

This usually means something like:

• should
• ought to
• be supposed to

So:

• ég á bók = I own a book
• ég á að lesa bókina = I should read the book

In your sentence, ég eigi að samþykkja boðið means I should accept the offer, not I own to accept the offer.

Because samþykkja is an infinitive here, and að is the normal infinitive marker, similar to English to.

• samþykkja = accept
• að samþykkja = to accept

After eiga in the sense of should / be supposed to, Icelandic uses að + infinitive:

• ég á að fara = I should go
• ég á að samþykkja boðið = I should accept the offer

So að belongs with the infinitive phrase.

Enn here means still.

So Ég er enn óviss... means I am still uncertain...

Its position is normal:

• Ég er enn óviss = I am still uncertain

In Icelandic, short adverbs like enn often come after the finite verb (er) and before the adjective or main idea they modify.

Compare:

• Ég er enn hér = I am still here
• Hann er enn veikur = He is still sick

Boðið is the definite form of boð, a neuter noun.

• boð = offer, invitation, proposal, bid
• boðið = the offer / the invitation / the proposal

In this sentence, it is the direct object of samþykkja:

• samþykkja boðið = accept the offer

Because boð is neuter singular, the definite form is boðið.

Also, for many neuter nouns in Icelandic, the nominative and accusative singular look the same, so you do not see a separate object ending here.

Because this is a subordinate clause, not a main clause.

In Icelandic main clauses, the finite verb often comes early because of V2 word order. But in subordinate clauses introduced by words like hvort, the word order is usually more straightforward:

• hvort ég eigi að samþykkja boðið

That is:

1. subordinator: hvort
2. subject: ég
3. finite verb: eigi
4. rest of the verb phrase: að samþykkja boðið

So the clause keeps the normal subordinate order, not main-clause inversion.

Óviss is an adjective meaning uncertain.

It is used as a predicate adjective after vera:

• Ég er óviss = I am uncertain

Then Icelandic adds um to show what the uncertainty is about:

• Ég er óviss um þetta = I am uncertain about this
• Ég er óviss um hvort... = I am uncertain about whether...

So the structure is:

• ég = subject
• er = am
• enn = still
• óviss = uncertain
• um hvort... = about whether...

Yes, that would also be natural.

The two are very close:

• óviss = uncertain
• ekki viss = not sure

So:

• Ég er enn óviss um hvort...
• Ég er enn ekki viss um hvort...

Both are fine. The first is a bit more compact; the second may sound a little closer to everyday English I’m still not sure whether...

The difference is small, and both are common and understandable.

It can mean several related things depending on context.

Common meanings of boð include:

• offer
• invitation
• proposal
• bid

So samþykkja boðið could mean:

• accept the offer
• accept the invitation

The exact meaning depends on the situation, not on the grammar. The sentence structure stays the same either way.