To tell you the truth, I do not know why that expression is included in your dialogue. Is there additional context that follows that might help us?
It appears that Mom is instructing Daughter on the use of a telephone and it's understood that they will be separated - perhaps far apart. It's possible that "Just save a place for me" may refer to the Mom's desire to visit her daughter should she call.As I said, I do not know for certain why that expression was used.
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