I am not a teacher.
Both 1. & 2. would be possible depending on the context, but 1. seems to be the more natural in the absence of additional information.
The sentence means, "I told her that I had lied to my mother (about the fact/when I said) that I needed money to buy some books."
If you needed money at the time then that is what you should say.
If you had needed money at the time then you say that.
There is no need for the past perfect just because it is used in the first part of the sentence. It is required only if it makes sense.