Some Precisions weigh in at 8lbs or less, Jazzes are generally more but I am sure there must be some lighter ones.
When it comes to specific lightweight basses, there is a Fender Precision Lyte, which is an active P/J with a very thin neck. About 7lbs. Doesn't quite sound like an 'old school' Precision or Jazz but is very nice.
There is also a rarer MB-4 which doesn't look like a Fender, passive P/J, sounds more like a Jazz, also with a skinny neck and around 7lbs.
I am still evangelistic about my newest squeeze - the Harley Benton PB Shorty. Weighs around 7lbs, sounds exactly like a Precision bass but is short scale. An impossible bargain at £80 new.