Creating ScriptableObject assets with custom names

[MichalOp]

I need to create an ScriptableObject asset with custom name (like when I create material, I can type the name).
However, this code allows me to only create assets with fixed name:

public static void CreateAsset<T>() where T : ScriptableObject
    {
        T asset = ScriptableObject.CreateInstance<T>();

        string path = AssetDatabase.GetAssetPath(Selection.activeObject);
        if (path == "")
        {
            path = "Assets";
        }
        else if (Path.GetExtension(path) != "")
        {
            path = path.Replace(Path.GetFileName(AssetDatabase.GetAssetPath(Selection.activeObject)), "");
        }

       string assetPathAndName = AssetDatabase.GenerateUniqueAssetPath(path + "/New " + typeof(T).ToString() + ".asset");
        AssetDatabase.
        AssetDatabase.CreateAsset(asset, assetPathAndName);

        AssetDatabase.SaveAssets();
        AssetDatabase.Refresh();
        EditorUtility.FocusProjectWindow();
        Selection.activeObject = asset;
    }
}

How could I make this name the name I write like when creating materials, scripts, folders etc?

[edit] I can change the name simply by clicking it, but if someone knows clean solution which allows me to make it behave like build in objects, it would still be apperciated.

[ninjapretzel]

@MichalOP Tagging the type with the CreateAssetMenu makes sure it shows up in the Create Asset Menu.

[CreateAssetMenu(fileName = "DefaultFileName", menuName = "Create Menu Display Name", order = 42)]
public class CustomAssetName : ScriptableObject {
    public string customDataFieldOne;
    public float[] customDataFieldTwo;
}

If you just want to rename existing assets, the only thing I can think of is to use

AssetDatabase.RenameAsset(string pathName, string newName);

This would work for your case if you use name data from a custom text field typed in by the user before invoking the CreateAsset method that you already have, and then modify CreateAsset to take in a name parameter and apply it to the created asset.