I have rarely lived with a pet and had no idea how to respond to the question of how to grieve Jewishly. I am grateful to those who shared these resources:
From Coffee Shop Rabbi — “May I Say Kaddish for My Pet?” — this post includes links to other resources.
Another rabbi answers “Is It Appropriate…?”
From Ritual Well: Grieving the Loss of a Pet
When a Beloved Pet Dies (PDF from Hebrew Union College)
Another friend suggested, “speaking as a pet parent who has lost pets,” that an appropriate response is a donation to a pet shelter.
Still another said that kaddish is a “public affirmation of belief in the power and sovereignty of the Almighty, even in times of grief,” so why not? It should be noted, however, that this suggestion is contrary to a number of prominent views about the recitation of Mourners’ Kaddish. See also this general post on Jews and Pets from My Jewish Learning.
General Resources on Kaddish
Many search results at My Jewish Learning
Audio lessons from Elie Kanfer at Hadar: “A New Interpretation”
An English-language kaddish for a not-necessarily-Jewish “minyan”