'Hatton' means the farmstead on the heath.
It comes from Anglo-Saxon Hǣþtūn, heath farmstead; until 1819 Hatton's fields and gardens were surrounded by heath. To the east lay Hounslow Heath, centred roughly where Heathrow's Terminal 3 is today. 'Cross' is less certain: probably the crossroads on the old coaching road to the south-west, now the large road junction just south-east of the station.