To me, a "peeve" is a complaint, and, here, "pet" means "favorite."

"Dan's pet peeve" is some specific thing that Dan actually seems to enjoy complaining about. 

"It's so hard to open packages nowadays."

"Oh, Dan, give it a rest. That's your pet peeve. You're always complaining about that. Stop it. I know you always carry a Swiss army knife with you. Use it to open packages or don't use it, but quit griping about it."

Do people really enjoy complaining? Of course they do. To quote the words of a Gilbert & Sullivan song:

Oh, don't the days seem lank and long,

When all goes right and nothing goes wrong?

And isn't your life extremely flat

With nothing whatever to grumble at!

I think that the best way to go about this is to turn "pet peeve" the subject of the sentence. For example "My pet peeve is when people eat and talk at the same time." (Meaning: I don't like it when other people eat and talk at the same time)

You can also do it like this (using your example): "My mother doesn't like it when the family watches TV during dinner. It's her pet peeve."

I hope this helped! ^_^

I believe that you are thinking about this far too much. A "pet peeve" is a personal dislike, and it's usually something insignificant - I mean, other people would not be bothered by the same thing  (thus the word "pet").

Whether you use "of" or "about", the pet peeve is still something you dislike and avoid.