pater liberos suos amat, the father loves his children.
[Essential Latin Grammar (by Anne Mahoney), page 49.]
Why is liberos=children?
I don’t really know why, but I can speculate:
My thought has always been that it is based on the perspective of the pater wielding patria potestas. From his view, he has two groups of dependents/property: his servi and his children. In order to distinguish his “non-free” property from his “free” property, this latter group is called liberi.
Pure speculation.
I think this is right (it’s the usual explanation given). A slave child born in the household is a verna. I am not sure if a distinction would be made for a slave child who would be purchased and not born in the household…