George Alexander Louis--a name replete with family history an connections.
George--the name of six previous Kings (including the Queen's beloved grandfather and the reigning name, if not the given name, of her father) as well as the name of the last Prince and Duke of Cambridge prior to William and baby George
Alexander--the name of the founders of both the Teck (Queen Mary) and Battenberg/Mountbatten (Phillip) houses as well as a nice nod to Scotland--much-loved by the British royal family and perhaps a nod to where the couple met
Louis--The Hessian line is well-represented here both through the lines of Ludwig IV and Princess Alice and through the Battenbergs (Louis Battenberg and Louis Mountbatten). Charles has the name George where William has the name Lous amongst their 4 names as well.
So there were nods to Elizabeth, Phillip, William and Charles (and the Cambridge name) in the name pick. It's a nice tying together as well of the various branches--Teck/Cambridge, Windsor, Hesse/Battenberg/Mountbatten which make up this generation of the family. A very royal name with a very royal heritage given the baby's position.