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Basically I have a tile which I then duplicated using an array modifier to get this:

Then in the node editor I created a procedural marble texture using nodes and got this:

The problem with this image is that it looks like the texture is spread across every tile, making it look unnatural. Is there a way to make the marble texture random for each tile?

Here is the blend file: http://pasteall.org/blend/index.php?id=50738

One way of doing this would be:

Separate all your tiles as a single object so you can use the Object info node random output in order the create a tile black and white mask:

You can then create another wave texture and combine it with the one you already have using the tile mask as factor:

You obviously need to change the texture coordinate of the second wave to create variation. Put a color ramp node and play with its value to have stronger difference:

http://pasteall.org/blend/index.php?id=50740

As @robin betts points out in the comments, if you need to keep the array modifier you can duplicate the object, do the same process, and bake the mask into a file image. Or if you want to keep a "semi" procedural creation use dupliface instead of the array.

You can try the brick texture to separate each tile. Here, I tried editing your file to get something that you wanted.

I was helping someone else with this issue a while ago so I'll be happy to elaborate.
To change the colours and almost every setting, ONLY use the encircled Brick node.

This setup will give you:

Simply change the colours in the circled brick node to suit your needs and the variation of colour and the patter colour will change. You can also try changing the distortion and scale values of the noise textures to suit your needs.
If you want the file here's the drive link: Link