Mother and son had been living in Bahrain since the boy was a toddler. I would presume he's been being called Adam for quite some time (probably ever since he's been with his mum). She even remarried before her ex died in the accident.

If the father and his family were concerned about the boy and how he was being raised, why wait until he was ten to do all this? He was living in Bahrain and presumably they also follow Sharia law, so would the authorities not have probably also awarded custody to the father/his family had they filed case there (as that is the way things are done per Sharia law)?
And from all reports I have read, mother and son had visited Qatar before -- so there were other, earlier chances to file suit (or abduct the boy, as they did in this instance).
Why now?

"If you're looking for sympathy, you'll find it between sh*t and syphilis in the dictionary."
- David Sedaris