You are likely looking for a Map. Maps allow you to identify values by a hashable key, such as a name string. Popular implementations include HashMap and TreeMap. The former is generally faster, while the latter is sorted:
Map<String, List<String>> mapOfLists = new HashMap<>();
Arraylist<String> stringListOne (lets say this has 3 elements)
Arraylist<String> stringListTwo (this one too)
mapOfLists.put("StringListOne", stringListOne);
mapOfLists.put("StringListTwo", stringListTwo);
You can now access your lists by name:
List<String> someList = mapOfLists.get("StringListTwo");
This is not exactly what you have in your example though. It is not advisable to use mutable objects for keys, since the hash value is likely to change. Instead, you may want to just map indices to names. You don't strictly need a Map object for this, since a List can be interpreted as a special case that maps contiguous non-negative integers to values. The easiest solution may be to create a list of names parallel to listOfArrays:
List<String> listOfNames = new ArrayList<>();
listOfNames.add("StringListOne");
listOfNames.add("StringListTwo");
Now you can do
String nameOfFirstList = listOfNames.get(0);
Map<String, List<String>>or create a wrapper object, etc.