If you're happy for the content type name to be saved within the "name" field, you could do this by hooking into the "SavingContent" event and prefixing the name before it saves like this:

[InitializableModule]
[ModuleDependency(typeof(EPiServer.Web.InitializationModule))]
public class BlockNamingInitialisation : IInitializableModule
{
    private IContentEvents _contentEvents;
    private IContentTypeRepository _contentTypeRepository;

    public void Initialize(InitializationEngine context)
    {
        _contentEvents = context.Locate.ContentEvents();
        _contentTypeRepository = context.Locate.ContentTypeRepository();

       //Attach the event handler to the creating and saving events
        _contentEvents.SavingContent += AddTypePrefixToBlock;
        _contentEvents.CreatingContent += AddTypePrefixToBlock;
    }

    private void AddTypePrefixToBlock(object sender, ContentEventArgs e)
    {
        //Only modify blocks whose names aren't already prefixed with a [
        if (e.Content is BlockData && !e.Content.Name.StartsWith('['))
        {
            //Fetch the content type name (you may want to cache this lookup - it'll get called a lot)
            var contentTypeName = _contentTypeRepository.Load(e.Content.ContentTypeID).LocalizedName;

            //Prefix the content name with the content type name
            e.Content.Name = $"[{contentTypeName}] {e.Content.Name}";
        }
    }

    public void Uninitialize(InitializationEngine context)
    {
        //Remove the event handlers on uninitialise
        _contentEvents.SavingContent -= AddTypePrefixToBlock;
        _contentEvents.CreatingContent -= AddTypePrefixToBlock;
    }
}