There are a few interpretations of what the title could mean: an overview, a bridge between characters, Catherine's bridge between girl and woman, a bridge between two worlds (America and Italy) and Eddie's change of character.
Alfieri is an outsider, his view is 'from the bridge'; he comments on Eddies' progress as a storyteller,
"I could have finished the whole story that afternoon...I could see every step coming." (p50)
This is what Alfieri says after Eddie comes to see him. He is the narrator and is quite similar to the chorus in a Greek Tragedy.
Characters also make 'bridges' or connections between other characters. For example, throughout the first act we see that Beatrice is the bridge between Catherine and Eddie and she still is on page 80 when she insists that Eddie attend the wedding when Catherine does not care. Also, Eddie dislikes Rodolpho and Catherine bridges that gap and tries to persuade Eddie to her point of view,
"What're you got against him?...He
only blesses you."(40)
Catherine's growing up is a bridge as well:
"I'm not a baby...Beatrice says to be a woman," (62)
In this scene she also asks Rodolpho to teach her and they venture towards the bedroom, she emerges to speak to Eddie adjusting her dress. An innocence of virginity is seen. The audience also sees a bridge in Catherine's job (18), with her growing up and changing from the rough area of the docks to something more 'up market'.
The title could also mean bridging the gap between two cultures, America and Italy. The Italian way of life is very much based on unwritten rules of honour and trust, the American justice system is written in stone,
"MARCO:...All the law is not in a book.
ALFIERI: Yes. In a book. There is...