Like you said in another post, it's the reflection of life and knowing your years have gone by "wasted," at least in the eyes of those who suicide on New Year's Day.
When the New Year rolls around, a lot of people become retrospective about their life and wonder if there's different things they could have done or if they've wasted it. I guess they look at it like, "Another year is another year closer to the grave - have I wasted my life?" and many would often find they have, even if they objectively haven't.
Suicide is also very common on Christmas Day, especially in prisons. This one makes sense because Christmas Day also makes people realise they've got nobody in life (if they haven't or think they haven't) and they know spending time alone on that specific day isn't normal, so it makes them think negative thoughts.
As for prisons - well, the inmates would rather spend Christmas Day with their loved ones over being holed up in a facility.
It's depressing to think about.