In Matthew's account of the Good News, Jesus's last words are:
"All power in heaven and earth have been given to me. Go therefore, and make disciples of all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Spirit; teach them to observe all that I have commanded you. And behold, I am with you always, until the end of the age."So making the Sign of the Cross is a renewal of our Baptismal promise.
To take on a name is to enter into a family. In the time of Christ, that engendered a degree of totality that is almost incomprehensible to us today. There were no social programs that served as safety nets. There were no Greyhound buses to California. There was no French Foreign Legion. Family was life.
The closest image I can give of the emotional impact of taking on a name is when the Bride and Groom are at the altar. The Groom presents the ring to the Bride and holds stationary in space. The Bride either willingly places it on her finger by pushing her finger through it. Or she opts to not enter into marriage and take the Groom's name.
When a Catholic (or other Christian) performs the Sign of the Cross and says those words, we are joyfully reaffirming our baptismal promises and acknowledging our privileges and responsibilities as brothers and sisters of Christ.