Body parts and possessives
From A Comprehensive French Grammar, 228-229,
use the indirect object pronoun to refer to the person affected when the action applies to someone else's body. The indirect object pronouns (complément objet indirect) are me, te, lui, nous, vous, leur.
Il m'a tordu le bras. He twisted my arm.
Elle lui lave les cheveux. She washes his hair.
Il lui a craché à la figure. He spat in his (or her) face.
This is the same construction for your example,
Je lui ai pris la main. I took her hand.
"I took her hand" is an action the subject applies to someone else's body and uses the COI/indirect object pronoun.
The meaning is different from the reflexive pronoun and les verbes pronominaux (me, te, se, nous, vous, se) other than that the reflexive is often used with body parts. The underlying grammar is similar, but the reflexive is used here when the subject and verb are referring to the same person's body. It functions like an indirect object in word order.
Vous vous êtes cassé le bras. You have broken your arm.
Elle s'est tourdu le bras. She wrenched her (own) arm.
"I saw her mother" does not relate an action to a body part of another person, so you would use the possessive adjective just like in English, "J'ai vu sa mère." To keep it simple, no body parts. See section 223. Mon/ma/mes, ton/ta/tes, son/sa/ses, notre/nos, votre/vos, leur/leurs are the possessive adjectives.
Related: Possessive adjective before a body part
French possessive adjectives vs reflexive pronouns