I have seen this term used in a couple of different ways recently.
One is to describe children born via embryo adoption. This is when by using in vitro fertilisation, an infertile couple can have a baby, neither of them being one of the donors. The snowflake part seems to come in because the embryos are frozen along the way.
The other is to refer to young people who are becoming adults in this decade. In some circles they are being described as the Generation Snowflake, with an inflated sense of their importance, overly quick to take offence or get upset (the mall is closed one day and it is a major disaster or the fast broadband service goes down and they burst into tears).
Apart from this low threshhold of emotional vulnerability they also display less resilience.
It is not a particularly favourable term and reminds me of the word ‘flaky’, a slang word for crazy, unreliable or wacky.
Jim’s new girlfriend is definitely flaky, you don’t know what she is going to come out with next.