Does memory persist across different births of an individual? I've read that the Buddha could remember his past life. Does this means that the memory of a person lasts across different life times?
This is just knowledge (jnāna or vidhyā). Anyone who acquired that knowledge can remember his past life. 

Please see Buddha's awakening Buddha's awakening

and page 63 of "On Being Buddha: The Classical Doctrine of Buddhahood"

and I retrieved the text below from page 571 of "Agama Aura Tripitaka: Eka Anusilana"



Prodigy of Memory of Earlier Births (Pϋrva-nivāsa-anusmrti-jnāna): The monk endowed with this prodigy remembers many kinds of earlier births. He can remember many periods of duration of dissolution, creation and destruction-cum-origination of the universe. He remembers that at that time, he was in a particular place, state of birth, species, destinity and state of knowledge. I was in a specific state of birth. I had this name, lineage, life-span, food and experience of pleasure and pain. I was born in a specific place after death from there where I had this…etc. After death there, I am born here in this Ksatriya or Brahmin family. The heretics remember their earlier births upto 40 Kalpas (a unit of time equivalent to 20 Antar-Kalpas). The natural votary remembers earlier births upto 100 or 1000 Kalpas, the great/senior votaries remember earlier births upto 80 lacs of Kalpa years, the self-enlightened one remembers the earlier births upto two innumerable lacs of Kalpa years and the Buddha remembers the earlier births upto a period without measure.