Learn Turkish: Aileme haber vereyim mi? - Shall I let my family know?

Vereyim is made from the verb root ver- (to give) plus the optative suffix -eyim (let me). Adding the separate question particle mi turns "let me give" into "shall I give?". This is the standard way to offer to do something or ask for instructions in Turkish.

Why is the word for "know" not used here to say "let know"?

In Turkish, the set phrase for "to let someone know" or "to inform" is haber vermek, which literally translates as "to give news". So in this sentence, you are literally asking, "Shall I give news?"

Let's break down the word aileme. Why does it end in -e?

The phrase haber vermek (to give news) requires the dative case to show who is receiving the news. Aile is family, ailem is my family, and the dative -e makes it aileme, meaning "to my family".

Could I just use the future tense here and say verecek miyim?

No, they mean different things. Asking haber verecek miyim? (Will I give news?) asks for a prediction about the future. To ask for someone's preference or permission, like "Shall I?", you must use the optative form vereyim mi?.