I could be way off base here - I haven't watched any Mindy eps for years and I don't have time to check one out now - but I have a memory in my head of it being a little, white, fluffy thing.
So my guess - unless my memory sucks, which is always possible - would be a Pomeranian
I thought she called the dog Pookie too in 'memory' of her husband or something. I didn't find 'dog' in the script at all [well twice but not in reference to Mindy's dog].
I got the name from HIWTHI she's in the car after phoning Joey Bermuda and framing him for the whole crime spree. She then picks up the dog (which I think is a French Poodle) and kisses the dog calling him 'Pookie'
Sounds like a fabulous plan. The next time I bring home a toy poodle, I'm going to dress him up in a girlie white sweater and name him "James." I'm sure it would do loads for my husband's ego.
Elisabeth who was trying to remember myself, but all I could think of was "Pookie" and I knew that was Bill