I think so, for a couple of reasons. For one thing, consider God's original creation. He filled the sky, seas, and land with animals (days 5 and 6). If the original creation, why not the new heavens and new earth? While there will be much discontinuity between the original and the new, the Scriptures give warrant to think there will also be a fair amount of continuity.
Further, animals not only filled the original creation, they also fill the pages of Scripture. And not just as the props of the various historical accounts (Saul's lost donkeys, the lion Samson killed), but often as illustrations, similes, and metaphors of spiritual truths. Proverbs would have us study the ant to learn industry. Aspects of both the lion and the lamb are gateways into understanding the Lord Jesus better. We are commanded to be as wise as serpents while at the same time being as innocent as doves. That God filled both creation and the Scriptures with animals seems to indicate some sort of love for all these sub-human creatures of his. Thus it is not unreasonable to think they might also populate eternity.
Finally, there seem to be specific indications of animals in the new heavens and the new earth, as for instance Isaiah 11:6-9:
6 The wolf will live with the lamb,
the leopard will lie down with the goat,
the calf and the lion and the yearling[a] together;
and a little child will lead them.
7 The cow will feed with the bear,
their young will lie down together,
and the lion will eat straw like the ox.
8 The infant will play near the hole of the cobra,
and the young child put his hand into the viper’s nest.
9 They will neither harm nor destroy
on all my holy mountain,
for the earth will be full of the knowledge of the Lord
as the waters cover the sea. (NIV 1984)
Will our specific pets be in heaven? That I don't know--I'm not aware of specific Scriptural teaching on that--but others have answered more definitely.