Other contributors have said "How and why did the old folk lore to be found under a gooseberry bush originate?" is the same question as "What does born under a gooseberry bush mean?" If you believe that these are not asking the same thing and should be answered differently, click here
I believe the phrase "born under" a sign refers to being born under your Ascendant (aka Rising Sign)--the sign that was ascending the eastern horizon at the exact time of birth.