Kotlin Firebase Realtime Database correct return value for setValue() in MVVM

What would be the correct return value of my function in the HomeRepository class for the code to work? Void doesn't seem to be the right return value.

Indeed Void is the type of object the setMyBirthday function returns. So your function should look like this:
//                                            👇
suspend fun setMyBirthday(birthday: String): Void =
databaseReference
    .child("users")
    .child(myUserID)
    .child("birthday")
    .setValue(birthday)
    .await()

While your approach will work, it will be more convenient to return a Resource<Boolean> as in the following lines of code:
suspend fun setMyBirthday(birthday: String): Resource<Boolean> = try {
    databaseReference
        .child("users")
        .child(myUserID)
        .child("birthday")
        .setValue(birthday)
        .await()
    Resource.Success(true)
} catch (e: Exception) {
    Resource.Error(e)
}

This is more useful because you can pass the error further, otherwise, you won't know if something fails.

What would be the correct return value of my setMyBirthday function in the HomeRepository class for the code to work? Void doesn't seem to be the right return value.
Fragment:
private fun HomeViewModel.setupObserver() {
    myBirthday.observe(viewLifecycleOwner) { response ->
        when(response) {
            is Resource.Error -> {
                response.message?.let { message ->
                    Log.e(TAG, "An error occurred: $message")
                }
            }
            is Resource.Loading -> {
                binding.buttonChangeBirthday.isClickable = false
            }
            is Resource.Success -> {
                binding.buttonChangeBirthday.isClickable = true
            }
        }
    }
}

HomeViewModel:
val myBirthday: MutableLiveData<Resource<Void>> = MutableLiveData()
fun setMyBirthday(birthday: String) = viewModelScope.launch {
    try {
        myBirthday.postValue(Resource.Loading())
        val response = homeRepository.setMyBirthday(birthday)
        myBirthday.postValue(Resource.Success(response))
    } catch (e: Exception) {
        myBirthday.postValue(Resource.Error(e.message!!))
    }
}

HomeRepository:
    suspend fun setMyBirthday(birthday: String) =
    databaseReference
        .child("users")
        .child(myUserID)
        .child("birthday")
        .setValue(birthday)
        .await()

