In England way back to probably the twelth century or earlier every village and town had a memorial cross, usually situated at its centre. In those days there were many Christian holidays and celebrations and the focal point for these celebrations was the village cross. Family names were not common, so men named John or Henry say, which was very common would be identified for example as John by the ford or the meadows, or the church, or in this case John by the "cross". Hence, John Crossby. Later evolving to Crosby or Crosbie which is probably the Scottish spelling.
(Sent from: Norman Campbell Crosbie)