Latinizing Lawrence gives it a whole new lease on life. Like Leonardo, Lorenzo has been integrated into the American stockpot of names, partly via actor Lorenzo Lamas. Other associations are with Lorenzo de' Medici, the Florentine Renaissance merchant prince and art patron, Renaissance artists Ghiberti and Lotto, and the upstanding young man who married Shylock's daughter Jessica in Shakespeare's The Merchant of Venice.
Lorenzo has been steadily in the 200-300 range on the popularity lists in the US, is currently Number 4 in its native Italy, and is also hot in France and the Netherlands. It would work well with a simple one-syllable Anglo name--in the depths of radio history, there was a soap opera character named Lorenzo Jones. And Nicole 'Snooki' Polizzi made this choice for her son.
Renzo and Enzo are both appealing nicknames, sometimes used on their own.