Stating a post doc research fellowship that I declined on my CV due to personal reasons? So, I have recently finished my PhD in one of the "pure" sciences, in the UK. I have applied and been successful in obtaining a prestigious and rather lucrative research fellowship for a post doc position at a university in a country abroad (different to UK). 

However, I have declined this award due to personal reasons, as I decided that a short-term post doc abroad without further career certainly is not something I want, due to my financial and family situation. Instead, I decided to either :

a) apply for a permanent position in the industry

b) apply for permanent teaching / lectureship positions in my current county of residence at "low" ranking universities / or so-called community colleges that emphasise teaching over research. (This is because I have virtually no chance of getting a permanent position without doing a post doc at a high quality research-orientated department.)

c) Apply only for post docs at "good" departments but only in the location I live now (since this way I will be able to save most of my salary, since I live at home with my parents), and transition to industry at a later date.

In either case, is it a good idea to state the fellowship that I have won, but declined, on either my CV and/or cover letter?
I don't think there is a clear answer to this that would apply in all situations. There are both positive and negative aspects to it. Yes, you were awarded a nice position. Yes, you backed out. For some situations one or the other of these might be more important. 

I'd suggest that for purposes of applying to a position you evaluate the situation and send a CV tailored to their needs. It may, in general, be safer to omit it if you think it might raise any questions. You can always bring it up in an interview later if you think it might be worth it. 

But most employers will want some assurance that backing out isn't a habit. So, for a general CV visible to the masses, maybe a bit better to omit it.