The most probable reasons they were refused are:
- not the right transit visa
- the transit visa maybe be not long enough between the flight from Europe into UK
- the job they are in (the qualifications may not be recognized by the US government)
- past jobs they have been in
- who they are or if they had any criminal past however petty
Sadly to say, our UK customs are very discriminatory and I feel they have no respect for UK passport holders or people born here even! The UK and US governments see people coming from countries that have been at war with them as very high risk passengers. They worry the passenger will abscond from the airport despite the large number that don't abscond. Here in the UK and the US more people somehow disappear on transit visas than on any type of visa.
My advice is make sure every angle is covered in getting a transit visa. Also, ask your Syrian relative to contact the UK and US embassies before leaving. The consulates can help to get right the paperwork for a transit visa for the UK and maybe any additional paperwork to convince both countries that your relative will return to Syria when stated. Also maybe take personal references from your relative's employers stating they work at [workplace] and what day they are expected to go back to work. Maybe even get a official letter from Syrian police authorities stating your relative has no criminal record.
I know its a pain. My advice is to be open and honest with either customs, to increase the chance you will get to see your relative. Also get every piece of paperwork checked at both embassies too, before your relative leaves, as both embassy have direct contact with customs and visa departments and they will without doubt tell your relative if it's ok to fly though the UK on to the US or not.