Are there any modern accounts (either of personal experiences of individuals on this forum, or in general) of an individual creating their own language and teaching it to their child(ren)/family?
If I remember right, the founder of Lojban used the language at home. I am not sure whether the children picked up the language.– Sir Cornflakes ♦May 13, 2018 at 18:15
One guy taught his daughter Volapuk.
There is a contemporary case of someone teaching their kids their personal conlang, she used to be active on one of Facebook's conlang groups.
The revived languages, Cornish, Hebrew, Sanskrit, have similar dynamics to conlangs. Having government support in the form of schools and day care works wonders for the project being successful.
Getting a child to speak your languages takes time (20+ hours a week) of exposure, and it really helps to have a community of peers who also speak that language, or a child's decision to keep using their parents language will be personal and idiosyncratic. I get this impression from people I've met who tried speaking Arabic or Icelandic to their kids in the US where they got plenty of exposure to their parents, but no particular supporting community of peers.
There's a pretty badly written account (English translation) of someone's children learning Arka, though I have no idea if it's true.
There's also the better-known case of d'Armond Speers teaching his kid Klingon.
Edit: the author of the first article later confirmed that it was a hoax.
1If this is a hoax, is there any value in keeping it here and linking to it?– curiousdannii ♦Jul 15, 2018 at 5:05
Not really, but I didn't know that it was when I posted this answer.– bb94Jul 17, 2018 at 3:19