Evolution of umbilical cord and reason of its elongation
The woman running bit is obviously a myth but that is genius.. Other than that the length of the umbilical cord has been found to be highly variable depending on each individual. Characterizing it as just long would therefore not be right. According to Wikipedia, 


  The umbilical cord in a full term neonate is usually about 50
  centimeters (20 in) long and about 2 centimeters (0.75 in) in diameter
  (reference).


Some complications during birth are 


  A number of abnormalities can affect the umbilical cord. The cord may
  be too long or too short. It may connect improperly to the placenta or
  become knotted or compressed (reference).


Why does it become too long has been associated with these factors.


  Long umbilical cords, defined as total length over 70 cm, have been
  significantly associated with:
  
  Maternal factors:
  
  Systemic diseases. Delivery complications. Increased maternal age.
  
  Fetal factors:
  
  Non-reassuring fetal status during labor. Respiratory distress. Vertex
  presentation Cord entanglement. Fetal anomalies. Male sex. Increased
  birth weight.
  
  Placental features:
  
  Increased placental weight. Right-twisted cords. Hyper-coiled cords.
  True knots (reference).


In conclusion, length of the umbilical cord has an average as mentioned before and being too long or short can lead to complications during birth. Regarding evolution of the cord, you could read this discussion.