Hello! Elin ; D
The original one 강아지 came to 멍멍이 then again 멍멍이 came a new word 댕댕이. You will see #댕댕이 a lot on Instagram which means it's a very good one to learn!
The change happened because 머 and 대 look very similar when you take a good look at them. You can see that they both look like a box. Hopefully you find my comment helpful.
Thank you so much for the kind explanation. I'ts very interesting story. ;)