Java method object name from argument

That's naming a variable - not an object. Objects don't have names, as such. (You can have a field called name of course, but it's not a general thing.)

Variable names are usually only relevant at compile time - so it doesn't really make sense to do this dynamically.

It sounds like you probably want a Map<String, YourClassType> so that you can associate each object with a name and then fetch it later. So something like:

class Foo {
    private final Map<String, Bar> bars = new HashMap<>();

    public Bar createBar(String name) {
        Bar bar = new Bar();
        bars.put(name, bar);
        return bar;
    }

    public void processBar(String name) {
        Bar bar = bars.get(name);
        if (bar == null) {
            // No Bar for that name...
            // Maybe throw an exception?
        }
        // Do something with bar
    }
}

You could use a Map where the key is your variable name and the value is ObjectOfSomeKind object:

public class YourClass {
  ...
  private Map<String, ObjectOfSomeKind> yourHashMap= new HashMap<String, ObjectOfSomeKind>();

  public method (string ARGUMENT1)
  {
    yourHashMap.put("someprefix_"+ARGUMENT1, new ObjectOfSomeKind ("someprefix_"+ARGUMENT1));

  }
  ...

  public retrieveMethod (string ARGUMENT1)
  {
    ObjectOfSomeKind objectOfSomeKindRetrieved = yourHashMap.get("someprefix_"+ARGUMENT1);
  }
}