In the midst of my previous argument about the gold standard, a number of libertarians advocated private money. Perhaps it would work. But there's a big empirical hurdle to overcome, notably, why doesn't it already exist? There's no reason that you couldn't start your own currency now; sure, you couldn't pay your taxes with it, but assuming a reasonable exchange rate, even that wouldn't be a problem.
We already have credible currencies with pretty strong competition between them. Why would anyone adopt yours? If you think you have a good answer, you should start a bank.