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A mutation is miscopying of heritable material. If you want to demonstrate that any such miscopying is random you're going to have to show that there was no cause.

I'm sorry Steve, but are you arguing in support of directed mutation?
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No

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Are you claiming that the changed sequence of DNA base pairs was actually caused in some way?
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Are you claiming that changes occur 'without cause' - i.e. 'randomly' (in the sense of being 'uncontrolled')?

No change without Cause - and we don't need the 'directed mutation' concept, which is an intellectual red herring in the context of your question - and there is still no need for magical sky fairies - the surviving children of the WWII Dutch Famine Winter had serious problems caused because of environmental factors - such as starvation.

It's now been understood that germ cell changes can occur if the parents experience certain environmental factors - then changes (miscopyings) DO occur (but not quite like the giraffes 'stretching their neck' to reach the top leaves!).

For example, the dietary efficiency of a child may be affected depending on whether the parents experienced severe 'feast or famine' or the young child did - the girls have their germ cells (ovaries) formed in vitro, but the guys germ cells (sperm) don't start getting getting formed till puberty. Consequently, if the 'hardships' are experienced during those particular times for each sex, then the research shows that genetic miscopyings occur. As a working hypothesis, there does seem to be a causal link that we as yet can't define exactly what chemical reactions occurred, what enzymes were in excess or shortage, etc.