Is urinary dribbling serious in a young girl or is it something that will go away as she matures?

Depends: What exactly do u mean by dribbling. Slow urination or leakage. Weak stream can be an anatomical problem such as stenosis. Leakage can be anatomical or just weak muscles. If just weak, kegal exercises could help with that

Urinary "dribbling": I would ask first what you mean by urinary "dribbling". Does she have wetness in her panties that could be just due to poor wiping. Does she hold her urine until the last second and then have some overflow as she tries to make it to the bathroom? Because "dribbling" as I would define it is not normal in a prepubertal girl. Also need to check for UTI, chronic constipation and other possible causes.

Evaluation by her : pediatrician for the issues described by Dr. Maes includes obtaining a clean-catch, first-voided (in AM, before eating or drinking) urine specimen. Other possible causes for " dribbling" range from "vaginal voiding" seen in some girls who are overweight (treated by having the child sit backwards on the toilet) to anxiety to behavioral responses to trauma. All should be checked out.