Now a number of states are expressed in French with the verb avoir "to have" plus a bare noun: avoir faim "to be hungry", avoir peur "to be afraid", avoir besoin "to need" etc. Given the preceding remarks, you would naturally assume that "need again" should be ravoir besoin - and, indeed, it is possible to find this expression at least in 19th century texts, eg:
Rentré dans le journalisme, cet esprit capable, mais aride et paresseux va ravoir besoin de moi. (1856)
It appears to be very little used in the 20th century, though. Instead we hear avoir rebesoin: j'ai rebesoin de ça, I need this again. The only Italian I asked said this is quite impossible in Italian, but even there ho ribisogno gets a few dozen hits on Google (though for all I know they're all second language speakers.)
The fact that besoin appears bare, with no article, already makes it unusual among nouns. The ability to take the prefix re- makes it stand out even more: you certainly can't say *revoiture (car again) or *repain (*bread again). So maybe it's not a noun any more? It certainly looks like it's become kind of verby; but what can we label it? In an Australian context, the uninflected element of a complex verb would be called a preverb, but apart from suggesting the wrong order of elements, this term has way too many different meanings depending on which part of the world you're in. Perhas, as in Japanese, we could call besoin a verbal noun - although that, too, is all too potentially ambiguous. Any better terminological suggestions are welcome.