Learn Hardcore German: Ihre Worte geben mir neue Hoffnung.

Because the word is at the start of the sentence, it’s capitalized anyway, so you can’t tell from capitalization alone. Only context will disambiguate it. If the sentence appeared mid-sentence as Ihre Worte, it clearly means formal your; as ihre Worte, it could be her or their.

Why is the verb geben plural and not gibt?

The subject is Ihre Worte (words), which is plural, so the verb agrees in number: geben. If the subject were singular, e.g. Ihr Wort, you’d have gibt: Ihr Wort gibt mir neue Hoffnung.

Why Worte and not Wörter?

German distinguishes two plurals of Wort:

Wörter = individual, countable words (dictionary entries, separate lexical items).
Worte = words as an utterance or statement, the content of what someone said.

Here, the idea is that what was said gives hope, so Worte is the idiomatic choice. Wörter would sound like you’re focusing on separate words, which is unusual in this context but not impossible.

Why is it mir and not mich?

Because geben takes a recipient in the dative case and a thing given in the accusative:

dative (to whom?) = mir (to me)
accusative (what?) = neue Hoffnung (new hope)

Using mich would make it accusative and ungrammatical here.

Could I say für mich instead of mir?

No. With verbs of giving (geben, schenken, zeigen, erzählen, etc.), German uses a dative recipient without a preposition: jemandem geben. Für mich means “for my benefit” and doesn’t mark the recipient properly here. Stick with mir.

What case is neue Hoffnung, and why?

It’s the direct object in the accusative case, because it’s the thing being given. The pattern is:

Subject (nominative): Ihre Worte
Verb: geben
Indirect object (dative): mir
Direct object (accusative): neue Hoffnung

Why is there no article before neue Hoffnung?

Abstract, uncountable-like nouns (e.g., Hoffnung, Geduld, Liebe) often appear without an article when speaking generally. You could add one:

Ihre Worte geben mir eine neue Hoffnung. This is also correct but slightly more concrete or individualized (“a new kind of hope” versus the general notion of new hope).

Why is it neue and not neuer or neuen?

Adjectives must take an ending that marks gender/number/case. Here there is no article before Hoffnung, so the adjective takes strong declension. Hoffnung is feminine singular accusative; the strong ending there is -e: neue Hoffnung. That’s why not neuer or neuen.

Why is mir placed before neue Hoffnung? Could I say Ihre Worte geben neue Hoffnung mir?

In the German middle field, unstressed pronouns typically come before full noun phrases. So:

Preferred: Ihre Worte geben mir neue Hoffnung.
Ihre Worte geben neue Hoffnung mir is technically possible but sounds odd and marked. You might front the dative for emphasis: Mir geben Ihre Worte neue Hoffnung, but that’s special emphasis or answer-focus.

Is the word order verb-second here?

Yes. In a main clause, the finite verb is in position 2. Ihre Worte occupies the first field as a single constituent, and geben is the second element:

[Ihre Worte] [geben] [mir neue Hoffnung].

Are nouns always capitalized in German? Why are Worte and Hoffnung capitalized?

Yes, all nouns are capitalized in standard German, so Worte and Hoffnung are capitalized. Pronouns like mir are lowercase. The formal Sie/Ihr/Ihre are capitalized wherever they appear.

How would I say this informally to one person?

Use the informal possessive:

Deine Worte geben mir neue Hoffnung. For several people informally:
Eure Worte geben mir neue Hoffnung.

If I mean her words, does the sentence change?

At the start of a sentence, it looks the same in writing: Ihre Worte geben mir neue Hoffnung. In the middle of a sentence, it would be lowercase for “her”: ihre Worte. Only context (and capitalization mid-sentence) disambiguates her/their vs. formal your.

Can I say Ihre Wörter geben mir neue Hoffnung?

Grammatically yes, but it’s less idiomatic. Wörter highlights discrete words; Worte is the natural choice for the idea that what someone said (their statement) gives hope.

Could I use machen or schenken instead of geben?

Yes, with nuances:

Ihre Worte machen mir neue Hoffnung. Very common and idiomatic (“make me hopeful again”).
Ihre Worte schenken mir neue Hoffnung. More elevated/poetic (“bestow new hope on me”). All three are correct; machen is the most everyday alternative to geben here.

Why is it Ihre Worte (nominative) and not Ihren Worten?

Because Ihre Worte is the subject (nominative plural). Ihren Worten would be dative plural (“to your words”) and appears after dative-governing prepositions or verbs, e.g. Mit Ihren Worten geben Sie mir neue Hoffnung.

Could Hoffnung be plural here?

You could say neue Hoffnungen, but it changes the meaning to multiple distinct hopes. The idiomatic expression is singular Hoffnung when speaking about the general feeling of renewed hope.

Is there any reason to prefer Wort (singular) over Worte (plural)?

If you mean a single utterance or promise, you can use singular:

Ihr Wort gibt mir neue Hoffnung. This subtly focuses on one statement or promise. With Worte, you attribute the effect to what she/you/they said overall.