One solution is to register a TypeDescriptionProvider for the string[] type. Here is the sample code (you need to register this at the beginning of your program, before displaying any property grid):

...
TypeDescriptor.AddProvider(new StringArrayDescriptionProvider(), typeof(string[]));
...

And here is the StringArrayDescriptionProvider code:

public class StringArrayDescriptionProvider : TypeDescriptionProvider
{
    private static TypeDescriptionProvider _baseProvider;

    static StringArrayDescriptionProvider()
    {
       // get default metadata
        _baseProvider = TypeDescriptor.GetProvider(typeof(string[]));
    }

    public override object CreateInstance(IServiceProvider provider, Type objectType, Type[] argTypes, object[] args)
    {
        // this is were we define create the instance
        // NB: .NET could do this IMHO...
        return Array.CreateInstance(typeof(string), 0);
    }

    public override IDictionary GetCache(object instance)
    {
        return _baseProvider.GetCache(instance);
    }

    public override ICustomTypeDescriptor GetExtendedTypeDescriptor(object instance)
    {
        return _baseProvider.GetExtendedTypeDescriptor(instance);
    }

    public override string GetFullComponentName(object component)
    {
        return _baseProvider.GetFullComponentName(component);
    }

    public override Type GetReflectionType(Type objectType, object instance)
    {
        return _baseProvider.GetReflectionType(objectType, instance);
    }

    public override Type GetRuntimeType(Type reflectionType)
    {
        return _baseProvider.GetRuntimeType(reflectionType);
    }

    public override ICustomTypeDescriptor GetTypeDescriptor(Type objectType, object instance)
    {
        return _baseProvider.GetTypeDescriptor(objectType, instance);
    }

    public override bool IsSupportedType(Type type)
    {
        return _baseProvider.IsSupportedType(type);
    }
}

This is how it will look: