Technical answer: if longevity is your priority, you need either to self-host everything or use a static blog generator. In the first case, you can install anything you want in your own server (Wordpress is always a good choice) and it will keep running as long as you pay your bills. For the second case (with something like Jekyll or Pelican), all you need is a website that hosts static files (namely, all of them) and if one goes down you simply re-upload somewhere else.
There's yet another point to consider: is there a reason why it has to be a blog? If you intend to give it to your child, a Word document is all you need.
Personal answer: I would encourage you not to blog in anger. I had to sort through a book written once by someone in pain for losing a child, and it was a very uncomfortable experience. I encourage to ask yourself "is this something that will make my child's life better, or is this something that will make my life better?". If it's the second, I would encourage you to still write it, but in a private diary rather than a public blog.