Můžete zkusit něco takového:
var objForUpdate = {};
if (req.body.nome) objForUpdate.nome = req.body.nome;
if (req.body.cognome) objForUpdate.cognome = req.body.cognome;
if (req.body.indirizzo) objForUpdate.indirizzo = req.body.indirizzo;
//before edit- There is no need for creating a new variable
//var setObj = { $set: objForUpdate }
objForUpdate = { $set: objForUpdate }
collection.update({_id:ObjectId(req.session.userID)}, objForUpdate )