If $R$ is a commutative ring, and $I\subset R$ is a non-zero free ideal, then it is principal generated by a non-zerodivisor.
If $R$ is a non-commutative ring having the IBN property and $I\subset R$ is a non-zero free ideal, then $I$ is not necessarily principal. A concrete example of a non-principal free ideal in $R=K\langle x,y\rangle$ is $I=Rx+Ry$. (Thanks to @rschwieb for pointed this out.)
For now I would like to find out an example of a non-principal free ideal in a (non-commutative) ring without the IBN property.
In order to do this I've thought to start with a ring $R$ with the property that $R\simeq R^3$ as (left) $R$-modules. Then the left submodule $I=\{0\}\times R\times R$ of $R^3$ is free (since $I\simeq R^2$), and therefore corresponds to a free left ideal in $R$. It remains to show that $I$ is not principal and here I'm stuck. If $R$ is an integral domain, then this follows immediately, but I don't know such an $R$. (In Lam's, Lectures on Modules and Rings, page 294, one can find an example of integral domain without the IBN property, but this doesn't satisfy the condition $R\simeq R^3$.)