Avia dicit omen bonum saepe augurium pacis esse.

Because after dicit (says), Latin normally uses an indirect statement construction.

That construction is:

• accusative subject
• • infinitive verb

So:

• Avia dicit ... esse = Grandmother says that ... is ...

In English, we usually use that + a normal finite verb:

• She says that it is ...

In Latin, that becomes:

• dicit ... esse

So esse is exactly what you would expect here.

Latin often leaves out a separate word for that in reported speech and uses the accusative + infinitive construction instead.

So English:

• Grandmother says that a good omen is often a sign of peace

becomes Latin:

• Avia dicit omen bonum saepe augurium pacis esse

The idea of that is built into the grammar of the infinitive clause, rather than being expressed by a separate word.

This is a very common point of confusion.

In an indirect statement, the subject is normally accusative, and a predicate noun with it is also accusative. So here, syntactically, both:

• omen bonum
• augurium pacis

are functioning in the accusative.

But they look like nominatives because they are neuter singular forms, and in Latin the nominative and accusative neuter singular are identical.

So:

• omen = nominative singular or accusative singular
• bonum = nominative singular neuter or accusative singular neuter
• augurium = nominative singular or accusative singular

So the grammar is accusative, even though the forms do not visibly change.

Yes, omen bonum is one noun phrase meaning a good omen.

The adjective bonum agrees with omen in:

• gender: neuter
• number: singular
• case: accusative here in function, though it looks the same as nominative

So:

• omen = omen / sign
• bonum = good

Together:

• omen bonum = a good omen

Pacis is genitive singular of pax, pacis (peace).

Here it means of peace:

• augurium pacis = a sign/augury of peace

In smoother English, that often becomes:

• a sign of peace
• or an omen of peace

So pacis depends on augurium and tells you what sort of sign it is.

Here saepe most naturally modifies the infinitive clause:

• omen bonum saepe augurium pacis esse
• that a good omen is often a sign of peace

So the meaning is:

• the omen is often a sign of peace

not primarily:

• Grandmother often says ...

If Latin wanted to make Grandmother often says especially clear, a very natural order would be:

• Avia saepe dicit ...

Because saepe is placed inside the reported statement here, it is most naturally understood with esse.

Latin word order is much freer than English word order. An infinitive like esse often comes at or near the end of its clause, especially in indirect statement.

So this order:

• Avia dicit omen bonum saepe augurium pacis esse

is perfectly natural.

Latin often arranges words for emphasis, rhythm, or clarity rather than following a fixed English-style order. Even though the order is different, the grammar still tells you how the words fit together.

Yes, they are close in meaning, and that is part of the point.

The sentence is equating one expression with another:

• omen bonum = a good omen
• augurium pacis = a sign/augury of peace

So the idea is:

• a good omen is often a sign of peace

Latin often uses near-synonyms like this in definitions or explanations.
There can be slight nuance:

• omen = omen, sign, portent
• augurium = augury, divinatory sign, omen

But in this sentence, the main function is simply to identify one thing as another.

Because Latin has no articles.

So Latin does not have separate words for:

• a / an
• the

That means omen bonum could mean, depending on context:

• a good omen
• the good omen
• sometimes just good omen in a more general sense

English has to choose an article, but Latin does not.

Avia is the subject of dicit.

It is:

• nominative singular
• feminine
• meaning grandmother

So:

• Avia dicit = Grandmother says

Everything after that is the content of what she says.

It can be translated either way, depending on context.

Latin present tense often covers both:

• simple present: says
• progressive present: is saying

So:

• Avia dicit could be Grandmother says
• or Grandmother is saying

In a sentence like this, English usually prefers says unless the context specifically emphasizes an action happening right now.