Writing a Factory method for STL random number generators

I'm trying to provide an interface — through a config file — for my users to choose a distribution for some of the parameters that they are using. I would like to use STL random number generator algorithms for this purpose.

Let's assume that my program reads a JSON from a command line. For the JSON provided below, the program needs to realize that it should generate a random number from the normal distribution with given mean and standard variation. (I'm using the same parameter names as STL library for clearance.)

{
    "dist": "normal_distribution",
    "mean": 0.1,
    "stddev": 0.5
}

So far, I can parse the JSON easily, and use each distribution's param_type to initialize the distribution. I use the name to decide which distribution to decide the param_type and the distribution.

What I don't know is how to implement this nicely. I know that I should provide some sort of factory method for this, pass the JSON, and spit out a function or a class. If I want to return an instance of a class, let's say a unique_ptr of the generator, I need to define an abstract class, e.g., RandDist and write some sort of adaptor to incorporate my input, .... I generally don't need a lot from the class, just a gen() method should be enough.

I'm wondering if anyone have thoughts on this. Or, if anyone knows a library that can do this.

P.S. Input doesn't have to be a JSON object, any hash table would work per se.