when(myMock.myFunction(anyString()).then(i -> i.getArgumentAt(0, String.class));

public class RoomService {
    public Room findByName(String roomName) {...}
    public void persist(Room room) {...}
}

RoomService roomService = mock(RoomService.class);
final Map<String, Room> roomMap = new HashMap<String, Room>();

// mock for method persist
doAnswer(new Answer<Void>() {
    @Override
    public Void answer(InvocationOnMock invocation) throws Throwable {
        Object[] arguments = invocation.getArguments();
        if (arguments != null && arguments.length > 0 && arguments[0] != null) {
            Room room = (Room) arguments[0];
            roomMap.put(room.getName(), room);
        }
        return null;
    }
}).when(roomService).persist(any(Room.class));

// mock for method findByName
when(roomService.findByName(anyString())).thenAnswer(new Answer<Room>() {
    @Override
    public Room answer(InvocationOnMock invocation) throws Throwable {
        Object[] arguments = invocation.getArguments();
        if (arguments != null && arguments.length > 0 && arguments[0] != null) {
            String key = (String) arguments[0];
            if (roomMap.containsKey(key)) {
                return roomMap.get(key);
            }
        }
        return null;
    }
});

String name ="room";
Room room = new Room(name);
roomService.persist(room);
assertThat(roomService.findByName(name), equalTo(room));
assertNull(roomService.findByName("none"));

public void testMyFunction() throws Exception {
    Application mock = mock(Application.class);
    when(mock.myFunction(anyString())).thenAnswer(
    invocation -> {
        Object[] args = invocation.getArguments();
        return args[0];
    });

    assertEquals("someString", mock.myFunction("someString"));
    assertEquals("anotherString", mock.myFunction("anotherString"));
}

public void testMyFunction() throws Exception {
    Application mock = mock(Application.class);
    when(mock.myFunction(anyString())).thenAnswer(
        invocation -> invocation.getArgument(0));

    assertEquals("someString", mock.myFunction("someString"));
    assertEquals("anotherString", mock.myFunction("anotherString"));
}

private Hashtable<InputObject,  OutputObject> table = new Hashtable<InputObject, OutputObject>();
table.put(input1, ouput1);
table.put(input2, ouput2);

...

when(mockObject.method(any(InputObject.class))).thenAnswer(
       new Answer<OutputObject>()
       {
           @Override
           public OutputObject answer(final InvocationOnMock invocation) throws Throwable
           {
               InputObject input = (InputObject) invocation.getArguments()[0];
               if (table.containsKey(input))
               {
                   return table.get(input);
               }
               else
               {
                   return null; // alternatively, you could throw an exception
               }
           }
       }
       );

import static org.mockito.AdditionalAnswers.returnsFirstArg;
import static org.mockito.ArgumentMatchers.any;
import static org.mockito.Mockito.when;

@Mock
private MyClass myClass;

// this will return anything you pass, but it's pretty unrealistic
when(myClass.myFunction(any())).then(returnsFirstArg());
// it is more"life-like" to accept only the right type
when(myClass.myFunction(any(ClassOfArgument.class))).then(returnsFirstArg());

public class MyClass {
    public ClassOfArgument myFunction(ClassOfArgument argument){
        return argument;
    }  
}

ArgumentCaptor<String> argument = ArgumentCaptor.forClass(String.class);
verify(mock).myFunction(argument.capture());
assertEquals("the expected value here", argument.getValue());