Congratulations on your forthcoming fatherhood. I'm not sure I can point to any product which is focused specifically on the "niche" you've described -- though

Congratulations on your forthcoming fatherhood. I'm not sure I can point to any product which is focused specifically on the "niche" you've described -- though maybe it's an opportunity because niche doesn't seem like the right word for such a huge market. I think most people today probably resort to whatever photo solution their phone offers them (either Apple Photos or Google Photos) and probably neglect to capture the less photographical memories of raising their children. But I think the key to any solution that you use is that it needs to be quick and easy. With that said, maybe something like a self-hosted or password-protected Wordpress installation would fit your use-case. It'd automatically chronograph your entries. It's got broad support for all types of media (photos, video, audio, etc.) and you can find limitless plugins to capture calendar entries and other memorable events. Just make sure you've got a rock-solid backup system in-place so that you don't risk losing all of your entries. If you want something more photo-focused, a self-hosted album like Koken might get you want you're looking for. It does provide the ability to blog as-well, though it's probably not as customizable as a Wordpress blog would be. Good luck with your newborn and finding a solution to help you remember all the little moments along the way.