It is not an uncommon occurrence in my family tree to find a person whose birth certificate contains a name they never used in their life. In my family tree, I have a set of triplets whose births were registered as Shadrach, Meshach, and Abednego. Within two weeks their parents saw reason and they were christened William, Albert, and Percy, the names which – believe it or not – they stuck with.
Technically I could describe them as William Jones né Shadrach Jones, etc. But what would be the utility of that? It would require further qualification and explanation, which would make the purpose of using né – its brevity – redundant. I entered them in my genealogy program under their christened names, which were used their entire lives. I made a note that interestingly their birth certificate names are different to the names they used. But a birth certificate is just a piece of paper, and as genealogists I think we sometimes set too much store by what that one piece of paper says. The concept of a birth name is in itself quite subjective – is it what the birth certificate says, is it what the mother first called the baby, etc.?
I would therefore reserve née and né for when the reason for its use is obvious, or explained elsewhere in your text. As other answers have stated, there is no reason it cannot be applied to given names, even though it may be less conventional to do so.
I would also add that in cases where you are using née to refer to changes in given names, it would be clearer to include the surname in both names even if it is the same surname. Rita née Rebecca Lerner implies that she was born Rebecca Lerner, but it does not make it clear whether she also used the surname Lerner later in life when she was called Rita. Rita Lerner née Rebecca Lerner leaves no ambiguity in this regard. If her married surname was Smith, you would be better writing Rita Smith née Rebecca Lerner.
In your mother's case, since she was known as Rita her entire life, I would hesitate to ever describe her as née Rebecca. Certainly it is a point worth noting that her birth certificate reads Rebecca. It could be a clerical error, and the parents never intended her birth certificate to say that. For all you know, Mrs Lerner could have sent Mr Lerner to register the birth, perhaps after he had had a glass or two in celebration of the new arrival, and his welcome when he got home: "I said Rita, you idiot! Not Rebecca!"