The Robin mantle is a method of reinvention at an individual level, and the Batgirl mantle is a study on independence at an individual level.
Both mantles, at a collective level, are symbols of hope.
But I also believe that both mantles are also the exploration of what it means to be someone's child.
Every Robin ever was someone's child. The triumph and tragedy of their Robinhood was that they were someone's.
Dick: created the Robin mantle as a memoir for his parents whose deaths created the fiery anger he molded himself into a better person with
Jason: was out on the streets because he was someone's son. died because he was someone's son. was resurrected because he was someone's son.
Tim: becoming Robin costed him a parent one too many times
Stephanie: was only Robin to spite her father, and along the way, forgot what it meant to be part of a family (then she'd give up her daughter later because of what Robin changed her life into)
Damian: became Robin because of the blood in his vein, raised to believe that the mantle was synonymous with familyhood for all the wrong reasons
Duke: became Robin because his home was in danger and his parents were stolen from him in the wreckage, stayed "Robin" (Signal) because it's all that might save them
It goes the same for the Batgirls
Barbara: became Batgirl to help her city and father, but all it did had worried her father to tears and made her a villain in her own home
Cassandra: used the Batgirl mantle to outgrow the sin of being the daughter of bad people, hoping the uniform would eventually absolve her of the blood in and on her
Even the Batman mantle exists and continued so because of a son's grief — Bruce's, Dick's (BotC), Damian's (future elseworlds).
The Batfamily is relevant to every character in it because the symbols and mantles exist because they are someone's child.