Thanks for the explanation. I'm trying to understand it per the rules given here: http://www.russianlessons.net/grammar/nouns_genitive.php but I can't
Oh boy... I'd be confused too, were I not a native speaker. In any case, "время" is a bit of an exceptional noun, which does not follow the standard pattern described there.
So, it's время (nominative and accusative), времени (genitive, dative, prepositional) and временем (instrumental).
I think this is an example of the temporal "for" use of на described on alphadictionary:
https://www.alphadictionary.com/rusgrammar/for.html (seach temporal)
I can see a scenario where it would work but it's not interchangeable with на and that's what you would use to express the usual meaning of this sentence, which can also be expressed as "...doesn't have time to spend on dogs", hence the на. With для she would be someone giving the dogs time like as a present. That's unlikely but theoretically possible I think.
У мамы нет времени на собак. | Mom does not have time for the dogs.
Mom doesn't have time for the dogs (I read this and think that there are dogs at home and someone else will deal with them).
Mom doesn't have time for dogs. (I read this as Mom has no time for any dogs and there are no dogs at home as a consequence)
What will be my contextual clue between these two meanings?