Ds's DNA bloods have been used at several hospitals so they may be able to use that sample.
Usually initial tests are for 'easy' to test for and rule out, syndromes, like Fragile X. Sometimes a specific syndrome is suggested by clinical history and appearance, and this can be specifically tested for, sometimes a wider net has to be thrown. The trouble is there are many, many syndromes, mosaicisms of syndromes, and many, as yet, undiscovered genetic conditions - so it can be difficult to suggest conditions unless there is something very obvious.
Many children (and adults) remain undiagnosed, even though it is almost certainly a genetic issue, this is because the science hasn't caught up yet and new syndromes are being discovered all the time as testing becomes more refined. undiagnosed.org.uk/
There are studies at the moment looking into genetic causes of developmental disorders www.ddduk.org/ The DDD study being the largest.
Both of the websites linked to have quite a lot of info about genetic testing.
I don't know if it is the same everywhere, but ds has a specialist genetics nurse as well as a geneticist - if you get one, ask for their email address, as they are great for answering questions.