I have never heard of a tradition that the mother could choose the
name of the first child and the father the second, or vice versa.
Among Eastern European Jews, most important was the custom that a
child not be named after a living relative - though two cousins might
both be named after the same grandfather, so duplication existed.
Generally, the first son was named after the paternal grandfather (if
he had died). In some families, the sons tended to be named after the
deceased relatives of the father and the girls after the deceased
relatives of the mother, in the order of grandparents first,
particularly revered ancestors next, then uncles and aunts of the
parents. In other families, all the children were named after the
deceased relatives of the father, leading to very clear naming
patterns detectable in successive generations. My experience is mainly
with Litvaks and the customs among Ukrainian or Romanian Jews or among
Chasidim might have been somewhat different.

JewishGen has some information about this at
https://www.jewishgen.org/InfoFiles/GivenNames/slide7.html and a few
subsequent slides. Both you and JewishGen refer to naming traditions
among "Ashkenazic Jews", but the Ashkenazim include German Jews, who
by the 19th century did not adhere strongly to naming traditions, and
a variety of Eastern and Central European Jews. Customs differed
somewhat >from region to region.

Judith Singer