Changing Water Texture to show Flowing Lava?

Are you using the latest version of Unity? You may not be seeing a change because of this bug http://forum.unity3d.com/threads/broken-water-basic-standard-assets.259671/#post-1724775

Try using the work around in that thread and then change the texture.

 

There is no diffuse texture for Unity water, the texture you replaced is a reflective colour cube which just reflects/refracts some colours from the cubemap input. As lava is not really reflective, refractive or transparent I would imagine you'd need to use a completely different material and a shader with at least diffuse texture inputs. Bumped Diffuse might be best, then you can animate the movement on the object by animating the UV's

Create new Material > Select Shader > Bumped Diffuse
Create Lava plane or select mesh > Animation window > Add curve > (create clip) Mesh Renderer > Material._MainTex_St > edit the position of the UV over time

Then you could do the same for a normal map (Material._Bump_ST) in another direction/speed to simulate rippling undercurrents and detail.