Which parent names the first child according to current custom?
Rav Ari Enkin in his Shulchan Ha'Ari page.133 writes that it is best to name the child together based off Koheles Rabbah 7:3. He notes that that some hold that the father gets to name the child first (Yabia Omer 5:21). Others hold that the mother gets to name first( Igros Moshe YD 3:101 and Keser Ephraim 39).
When we asked the Rov (Rav Y. Salzer shlita) about this (over 3 decades ago) he said that the mother gets to choose the first child's name, and thereafter alternate (father 2nd, mother 3rd, etc.)
IIRC he based this on Leah giving Reuven his name.
(Even though she gave Shimon his name too and Yakov (possibly - see Rashi) give Levi his name.)