Learn Spanish (Spain): Mi madre lee el horóscopo y dice que tu signo tiene buena suerte hoy. - My mother reads the horoscope and says that your sign has good luck today.

Because the reporting verb, dice (says), is in the present tense. When you report what someone says using a present tense verb, the tense of the original statement does not change. The mother's original thought used the present tense, so it stays as tiene in the reported sentence.

Can I leave out the word que after dice, just like we often drop 'that' in English?

No, in Spanish you can never skip que when connecting a reporting verb like dice to the reported statement. You must always use dice que (says that), even though 'says your sign...' sounds perfectly fine without 'that' in English.

How did the possessive adjective change from the mother's original words?

In reported speech, pronouns and possessives shift to match the current speaker's perspective. The mother probably said something like, 'El signo de tu amigo tiene buena suerte' or used your name. The speaker is now talking directly to you, so they logically change it to tu signo (your sign).

Why does the sentence use tiene buena suerte instead of a verb meaning 'to be lucky'?

In Spanish, luck is something you 'have' rather than a state you 'are' in. The phrase tener buena suerte literally translates to 'to have good luck', which is the standard Spanish way to express being lucky.