Learn Russian: Тёть, ты можешь перезвонить Диме? Он ждёт. - Auntie, can you call Dima back? He is waiting.

Тёть is the colloquial truncated vocative of тётя (aunt). To form this short vocative, you drop the final vowel to get someone's attention quickly. It is highly informal, so you should only use it with family members or people you are close enough to address with ты.

When we drop the final vowel from тётя, why does it become тёть with a soft sign instead of just тёт?

The vowel я in тётя indicates that the preceding consonant т is soft. When you drop the vowel to form the truncated vocative, you still need to preserve that soft pronunciation. The soft sign (ь) is added to show that the final т remains soft.

Why is Dima's name in the dative case (Диме) instead of the accusative?

Unlike in English where you call someone as a direct object, Russian verbs for telephoning like звонить and перезвонить require the dative case. You are essentially directing the call to him. Since Дима is a masculine name ending in -а, it declines like a feminine noun, making its dative form Диме.

Could I use the imperfective verb перезванивать here instead of the perfective перезвонить?