The correct answer is Cloud Functions. This GCP service is tailored for managing and automating the deployment of serverless functions, enabling developers to write and deploy event-driven functions without the need to manage the underlying infrastructure. It supports various programming languages such as JavaScript, Python, and Go, and functions can be triggered by events from GCP services or custom events.

• Cloud Build is a CI/CD platform in GCP focused on building and deploying applications, not specifically for serverless functions.
• Cloud Run allows running containerized applications in a serverless manner but isn't specifically designed for serverless functions.
• Google Kubernetes Engine (GKE) is a managed Kubernetes service for containerized applications, including serverless functions packaged as containers, but it's not specialized for serverless function deployment.

Thus, Cloud Functions is the correct choice for managing and automating serverless function deployments.