Hubo algún progreso hacia el problema anterior. Ahora hay un nuevo problema. El texto en el GridView muestra el resultado correcto. Sin embargo, las imágenes son las mismas que al comienzo de la lista.

Por ejemplo: si busco "Sidd"muestra tres resultados pero las fotos todavía comienzan como para los usuarios que comienzan con"A". Se adjunta una captura de pantalla para mayor claridad.

Este es el código BaseAdapter:

public class TagFriendsAdapter extends BaseAdapter implements Filterable {

    List<String> arrayListNames;
    List<String> mOriginalNames;

    List<String> arrayPictures;
    List<String> mOriginalPictures;

    Activity activity;
    String[] fetFriendID;
    String[] fetFriendName;
    String[] fetFriendPicture;

    LayoutInflater inflater = null;
    ImageLoader imageLoader;

    TagFriendsAdapter(Activity a, String[] stringUID, String[] stringName, String[] stringPicture,
            ArrayList<String> arrayName, ArrayList<String> arrayPicture) {

        activity = a;
        fetFriendID = stringUID;
        fetFriendName = stringName;
        fetFriendPicture = stringPicture;

        this.arrayListNames = arrayName;
        this.arrayPictures = arrayPicture;

        inflater = (LayoutInflater)activity.getSystemService(Context.LAYOUT_INFLATER_SERVICE);
        imageLoader = new ImageLoader(activity.getApplicationContext());
    }

    public int getCount() {
        return arrayListNames.size();
    }

    public Object getItem(int position) {
        return position;
    }

    public long getItemId(int position) {
        return position;
    }

    @Override
    public void notifyDataSetChanged() {
        super.notifyDataSetChanged();
    }

    public View getView(final int position, View convertView, ViewGroup parent) {
        View vi = convertView;
        if(convertView == null)
            vi = inflater.inflate(R.layout.friends_grid_items, null);

        ImageView imgProfilePicture = (ImageView)vi.findViewById(R.id.imgProfilePicture);
        TextView txtUserName = (TextView)vi.findViewById(R.id.txtUserName);

        txtUserName.setText(arrayListNames.get(position));

        if (arrayPictures.get(position) != null){
            imageLoader.DisplayImage(arrayPictures.get(position), imgProfilePicture);
        }
        else if (arrayPictures.get(position) == null) {
            imgProfilePicture.setVisibility(View.GONE);
        }

        return vi;
    }

    @Override
    public Filter getFilter() {

        Filter filter = new Filter() {

            @SuppressWarnings("unchecked")
            @Override
            protected void publishResults(CharSequence constraint, FilterResults results) {

                arrayListNames = (List<String>) results.values;
                notifyDataSetChanged();
            }

            @Override
            protected FilterResults performFiltering(CharSequence constraint) {

                FilterResults results = new FilterResults();
                ArrayList<String> FilteredArrayNames = new ArrayList<String>();

                if (mOriginalNames == null && mOriginalPictures == null)    {
                    mOriginalNames = new ArrayList<String>(arrayListNames);
                    mOriginalPictures = new ArrayList<String>(arrayPictures);
                }
                if (constraint == null || constraint.length() == 0) {
                    results.count = mOriginalNames.size();
                    results.values = mOriginalNames;
                } else {
                    constraint = constraint.toString().toLowerCase();
                    for (int i = 0; i < mOriginalNames.size(); i++) {
                        String dataNames = mOriginalNames.get(i);
                        if (dataNames.toLowerCase().startsWith(constraint.toString()))  {
                            FilteredArrayNames.add(dataNames);
                        }
                    }

                    results.count = FilteredArrayNames.size();
                    System.out.println(results.count);

                    results.values = FilteredArrayNames;
                    Log.e("VALUES", results.values.toString());
                }

                return results;
            }
        };

        return filter;
    }

}

Y la captura de pantalla: