I am aware that this is the right of the parents to name their newborn (children). (It could even be counted as a wajib/mandatory practice for parents to choose an appropriate and Islamic name for their children.) But I was wondering if it is considered as the right of father or mother to choose the name of their baby(s)?
Well even if I would agree with the answer of @Farhan totally, I must add that in the case that parents couldn't come to a mutual consent (at least Sunni) scholars tend to give this right to the father because of the part of the Verse in Surat al-Ahzaab (33:5) saying
Call them by [the names of] their fathers; it is more just in the sight of Allah .
This is interpreted as follows: As the child would have the (family) name of the father, so in case that the parents couldn't agree about the name of the child, the father would be the one who has more rights to chose (Fatwa in Arabic). See also this detailed fatwa about naming according to Sunni (more likely Salafi) point of view!
And Allah knows best!
To supplement the answers already given, evidence for the mother having the right to name the child includes the Quranic story of the mother of Maryam عليه السلام:
وإني سميتها مريم
I have named her Mary
ويدل أيضا على أن للأم تسمية ولدها ؛ وتكون تسمية صحيحة؛ وإن لم يسمه الأب; لأنها قالت: وإني سميتها مريم ؛ وأثبت الله (تعالى) لولدها هذا الاسم
And this is evidence for a child being named by the mother, and that the name she gives is correct, even if it has not been given by the father, for it is said: "I have named her Mary" and Allah retained the name given by the mother.
As you know, in Islam, there are many emphases and advices about choosing a good name for a child. For this reason, Islam considers the choice of a good name as the duty of parents and their first benevolence. The Messenger of God (pbuh) says in this regard: "The child has three rights over the father: first, that he chooses a good name for him, second, that he teaches him to read and write, and third, that he chooses a wife for him."
One of the children's religious rights, which is the responsibility of the father, is to choose a good name for them. In Islamic traditions, this important matter has been emphasized a lot. That is, if the child does not have a good name, the father will be reprimanded, and this does not mean that no one but the father has the right to participate in the naming.