The name Renée is indeed French, and literally means "reborn" (it is the feminine equivalent of René, which was Descartes' first name), ultimately descending from a Latin form renāta. It is the feminine form of the past participle of the verb renaître "to be reborn". Like many other French past participles, it has the ending -é (giving rené) and then declines like an ordinary adjective, taking an additional (unpronounced) -e in the feminine singular (hence renée). In this respect it behaves equivalently to another French past participle borrowed into English, fiancé & fiancée, the masculine & feminine singular past participles of the verb fiancer "to get engaged"
From now on though we're dealing with your specific spelling which is necessarily pretty much pure speculation
The macron on the first e is not used in French or English so is likely completely idiosyncratic. It also is not present in the Latin renātus with has a short e. Besides, the use of macrons for Latin is a relatively recent invention, Classical Latin generally lacked explicit marking of vowel length and, when it was included, used an apex, ancestral to the modern acute accent, rather than a macron. This is certainly idiosyncratic
The capitalised N is the only one that might have some linguistic basis, although I suspect it's very unlikely. The verb renaître is etymologically composed of two parts, re- "re-" and naître "to be born". There is a small chance that someone with some knowledge of the name's etymology was attempting to show this composition. This seems unlikely though as it's never been part of French or English spelling to mark something like this this way, and would require decent knowledge of the name's etymology, but the other oddities suggest a lack of such knowledge. If it is based on the etymology it must be earlier than the other changes, as someone making this distinction would doubtless have corrected the other idiosyncrasies. Most likely this is also purely idiosyncratic
Swapping the order of the é & the e would not make sense in French grammar or spelling so it's not really possible to say how it would affect the pronunciation. This may be a case of someone without any knowledge of French misreading, or misremembering the spelling, and the exchange getting fossilised, or may be deliberate idiosyncrasy. If it weren't for the other changes I'd lean in favour of the misreading, but with the others I'm inclined towards it being another idiosyncrasy
So two of the changes have possible (albeit in one case extremely unlikely) explanations beyond idiosyncrasy whilst the other has none. I suspect the most likely explanation is that some eccentric in your family history decided to spell the name in this unusual way for no particular reason, and it happened to stick