Η αγάπη της γιαγιάς μου μου δίνει κουράγιο κάθε μέρα.

Yes:

• Η — the feminine singular definite article, the
• αγάπη — love; this is the subject of the sentence
• της γιαγιάς μου — of my grandmother / my grandmother’s
  • της — genitive singular article
  • γιαγιάς — genitive singular of γιαγιά (grandmother)
  • μου — my
• μου — to me
• δίνει — gives; 3rd person singular present of δίνω
• κουράγιο — courage, strength, heart
• κάθε μέρα — every day

So the structure is basically:

[The love of my grandmother] [to me gives] [courage] [every day].

Greek uses the definite article much more often than English does. It is very normal to say η αγάπη even when English might simply say love.

So in Greek:

• Η αγάπη της γιαγιάς μου... is natural
• English may translate it as either My grandmother’s love... or The love of my grandmother...

The article does not always need to be translated as the in English.

Because Greek uses the genitive case to show possession or close relationship.

• η γιαγιά μου = my grandmother
• της γιαγιάς μου = of my grandmother / my grandmother’s

So:

• η αγάπη της γιαγιάς μου = my grandmother’s love

This is one of the main jobs of the genitive in Modern Greek.

Because Greek possessive words like μου, σου, του, της, μας, σας, τους usually come after the noun.

For example:

• η μητέρα μου = my mother
• ο φίλος σου = your friend
• το σπίτι μας = our house

So γιαγιάς μου literally looks like grandmother of-me, but in natural English it means my grandmother.

They do two different jobs:

1. της γιαγιάς μου
   Here μου means my.

2. μου δίνει
   Here μου means to me.

So the first μου is possessive, and the second μου is an indirect object pronoun.

This is very common in Greek.

In Modern Greek, the weak pronouns used for indirect objects are forms like:

• μου = to me
• σου = to you
• του / της = to him / her
• μας = to us
• σας = to you (plural/polite)
• τους = to them

So with δίνω (I give):

• μου δίνει = he/she/it gives me
• σου δίνει = gives you
• μας δίνει = gives us

Even though learners often first meet μου as my, it also regularly means to me.

Because the subject is η αγάπη, which is third person singular.

The dictionary form is:

• δίνω = I give

Present tense:

• δίνω = I give
• δίνεις = you give
• δίνει = he/she/it gives

Since love is a singular thing, Greek uses δίνει:

• Η αγάπη ... δίνει = Love ... gives

Because weak object pronouns in Greek normally come before a finite verb in ordinary statements.

So Greek says:

• μου δίνει = gives me
• σου λέω = I tell you
• τον βλέπω = I see him

This is different from English word order.

A common exception is the affirmative imperative, where the pronoun usually comes after:

• Δώσε μου = Give me

But in a normal statement like this one, μου δίνει is the expected order.

Because κουράγιο is being used in a general, abstract sense: courage / strength / encouragement.

After verbs like δίνω, Greek often uses abstract nouns without the article:

• δίνει κουράγιο = gives courage
• δίνει ελπίδα = gives hope
• δίνει χαρά = gives joy

If you added an article, it would usually sound more specific:

• δίνει το κουράγιο = gives the courage

That is possible only in a special context, not as the neutral meaning here.

Κουράγιο is a very common Greek word meaning something like:

• courage
• strength
• heart
• emotional strength
• encouragement

In this sentence, it does not necessarily mean bravery in battle-style English courage. It can also mean the inner strength to keep going.

So μου δίνει κουράγιο is a very natural Greek expression.

Because κάθε (each / every) normally goes directly with a singular noun without an article.

So Greek says:

• κάθε μέρα = every day
• κάθε χρόνο = every year
• κάθε φορά = every time

You do not normally say κάθε τη μέρα here.

Also, μέρα is the common everyday form of ημέρα.

Yes. Greek word order is more flexible than English, because the endings and pronouns already show the grammatical relationships.

This sentence can stay as:

• Η αγάπη της γιαγιάς μου μου δίνει κουράγιο κάθε μέρα.

But you could also say:

• Κάθε μέρα η αγάπη της γιαγιάς μου μου δίνει κουράγιο.

That puts more emphasis on every day.

What usually does not change in a neutral statement is the position of the weak pronoun before the verb:

• μου δίνει, not normally δίνει μου

In theory, a phrase like this can sometimes be ambiguous, but in ordinary Greek this wording is most naturally understood as:

• my grandmother’s love

If you want to clearly say love for my grandmother, Greek would usually prefer:

• η αγάπη για τη γιαγιά μου

So in your sentence, native speakers will normally understand the love that my grandmother gives/has, not love directed toward my grandmother.