My mother-in-law (US citizen) was refused a UK visa to visit her grandson. What are my options?

I am a British Citizen and my wife is American and has been living and working here in the UK with Indefinite Leave to Remain since 2007. Her Mother, who is the only family she has, visits regularly from the US to see her and our 2 year old son.

Last year she had a real hard time convincing border control to let her in, and she did stay slightly over 6 months that time. So when she got back home, she applied for a visa. It was denied for the following reason:

I am not satisfied you are seeking entry to the UK in the spirit of conditions of a genuine visitor.

She has no intention of trying to live here in the UK. With her application was a LOT of evidence showing that she only wants to visit for around 3 or 4 months a year to spend time with her daughter and grandson. She has tens of thousands in savings, owns her own home in the US debt free, is a Pastor in her local community etc... She stays with us while she is here and is no burden on the tax payer.

Is there anything else she can do to prove that she is not trying to live here? Also can she still visit even though her visa was denied, or will she be turned away at the border?

Edit to add scan of refusal: