Learn Icelandic: Ég er með bréfið frá barnabarni mínu.

It is because of the preposition frá, which always forces the words that follow it into the dative case. Since barnabarn is a neuter word, we need the neuter dative form of the possessive pronoun minn, which is mínu.

Why is there an -i at the end of barnabarni?

The base word is barnabarn (grandchild), which is a neuter noun. In the dative case (triggered by frá), most neuter nouns that end in a consonant add an -i to the end.

Why say Ég er með instead of using the verb að eiga to mean 'I have'?

While að eiga means 'to own', að vera með (literally 'to be with') is the standard way to say you physically have an object on you or in your hands right now. Since you are currently holding or carrying the letter, ég er með is the natural choice.

If frá takes the dative, why isn't bréfið in the dative case too?

The word bréfið belongs to the first part of the sentence with með (with). When með means having or holding something, it takes the accusative case. The preposition frá only controls the case of the words that come directly after it, which is the phrase barnabarni mínu.