Upgrading to the latest supported stable PHP branch is the recommended route plain and simple.
That being said, you've noted that you're with GoDaddy. Which of their services are you using for the project? If it's their shared hosting line, or any line that offers cPanel, upgrading PHP versions shouldn't add anything to current costs. Any web hosting provider with common sense should be able to easily run EasyApache, cPanel's built-in utility for updating and upgrading Apache, PHP and various other pieces of software.
If you're on a VPS, Cloud Server or Dedicated Server, the above still applies if you're using cPanel. If you're running barebones and someone is managing everything from the CLI (nice!), upgrading is a bit more time consuming, though not by much. They'll need to make sure the correct packages are installed before compiling, or, they could simply use built-in packages offered by upstream.
Even if you rely on certain packages, chances are upstream repositories can cater to your needs. If not, there's always building on to an existing package.
That said, if your boss is dead-set on sticking with dated releases, when it comes to PHP 5.3.7+, I'll reference ircmaxwell and paragonie, both of which are available on GitHub.
password_compat - link
The purpose of this library is to "...provide forward compatibility with the password_ functions that ship with PHP 5.5."
random_compat - link
The purpose of this library is to provide PHP 5.x support for
random_int() (from PHP 7.x). This isn't an alternative to password_compat, it's simply another compatibility layer that will allow you to bring features that you're not going to be able to use (unless you're able to upgrade, of course) down.
Note: If you're going to use random_compat, make sure you're using a stable release instead of the master (as noted by the author).