What is the ruling on my brother's work in programming songs for callers during the waiting period, and is it obligatory for him to leave this job and search for another, or should he continue in it until he finds a permissible job?

If your brother's job at the institution involves programming forbidden songs, then it is not permissible for him to continue, as it falls under aiding in the spread of evil and corruption. Allah Almighty said: "And cooperate in righteousness and piety, but do not cooperate in sin and aggression." If he fears Allah and leaves this work, Allah may replace it with something better for him. However, if there is a necessity for him to continue, then there is no harm, and necessity is determined by its extent. Once the necessity ceases, he must leave it.