I have never seen the Clone spell in use, but IIRC when the "original" dies, his soul goes into the clone which is then the same person as before but in a new body - I don't remember which are the penatlies for the change (level loss or anything).
If that's what the spell does, I'd definitely allow the new character to keep the familiar.
I'd say he keeps the familiar benefits, since his soul transfers to the new body in 3.0/3.5. Now, if it was 1st edition-style Clone, which basically created an identical but separate creature, then no.