Насколько я понимаю, вы сможете сделать это довольно просто, используя общие настройки.
Затем при запуске получить int и перевернуть фон
SharedPreferences preferences;
SharedPreferences.Editor editor;
@Override
protected void onCreate(bundle savedInstanceState){
super.onCreate(savedInstanceState);
setContentView(R.layout.myactivity);
preferences = PreferenceManager.getDefaultSharedPreferences(getApplicationContext());
editor = preferences.edit();
//I used a separate method to keep clutter out of onCreate but its up to you, doesn't matter.
setBackground();
button.setOnClickListener(new View.OnClickListener() {
@Override
public void onClick(View v) {
setBackground();
}
});
}
//Here we just grab the id, and assign the background.
public void setBackground(){
int id = preferences.getInt("backgroundid",1);
int bg = 0;
if (id == 1) bg = R.drawable.bg1;
if (id == 2) bg = R.drawable.bg2;
if (id == 3) bg = R.drawable.bg3;
if (id == 4) bg = R.drawable.bg4;
if (id == 5) bg = R.drawable.bg5;
layout.setBackgroundResource(bg);
editor.putInt("backgroundid",id++);
editor.commit();
}