Our beloved Yorkshire Terrier has died, due to complications of old age. What prayer(s) may be said, and, is there any one prayer, in particular, that stands out as the most commonly cited prayer upon the death of a pet dog?
חמורו שמתו " , המקום ימלא לך חסרונך ," כך אומרים לו על עבדו ועל שפחתו שמתו " , המקום ימלא לך חסרונך "
If someone's donkey died, they tell him "Hashem should fill your loss", like on a dead servant or dead maid servant. I suppose a dead dog is the same. I think the gemoro means one doesnt say a prayer or hesped on a dead servant only a well wish like on an animal which I am sure includes a dog. I think that is a direct answer on the question although it may not be what you want to hear.
I'm sorry for your loss, but I'd like to tell you a story. A friend of mine was raised on a farm. One of the horses grew old and was unable to work, but having been part of the family so long, the horse was kept around. My friend grew close to the horse, befriending it. Ultimately, the horse passed on and my friend, who at the time was still young was inconsolable. His grandfather took him aside, praised him for his commitment and caring nature, but told him 'We don't cry over a horse'. My friend says this was a very important learning experience that helped mold him.
There are no prayers for the loss of a pet, but there is place for introspection. In Ecclesiastes chapter 3 vs 18-19, king Solomon points out the seemingly similar end for both beast and man. Rashi in his commentary points explains Solomon's intention as the reason man and beast share a similar fate called death is meant to humble man so as not to be haughty, for ultimately everything is in the hands of God.
אָמַ֤רְתִּֽי אֲנִי֙ בְּלִבִּ֔י עַל־דִּבְרַת֙ בְּנֵ֣י הָאָדָ֔ם לְבָרָ֖ם הָאֱלֹהִ֑ים וְלִרְא֕וֹת שְׁהֶם־בְּהֵמָ֥ה הֵ֖מָּה לָהֶֽם׃ כִּי֩ מִקְרֶ֨ה בְֽנֵי־הָאָדָ֜ם וּמִקְרֶ֣ה הַבְּהֵמָ֗ה וּמִקְרֶ֤ה אֶחָד֙ לָהֶ֔ם כְּמ֥וֹת זֶה֙ כֵּ֣ן מ֣וֹת זֶ֔ה וְר֥וּחַ אֶחָ֖ד לַכֹּ֑ל וּמוֹתַ֨ר הָאָדָ֤ם מִן־הַבְּהֵמָה֙ אָ֔יִן כִּ֥י הַכֹּ֖ל הָֽבֶל׃
I said in my heart: ‘It is because of the sons of men, that God may sift them, and that they may see that they themselves are but as beasts.’ For that which befalleth the sons of men befalleth beasts; even one thing befalleth them; as the one dieth, so dieth the other; yea, they have all one breath; so that man hath no pre-eminence above a beast; for all is vanity.
Thank G-d for the happiness you had from having a pet. Thank G-d for the merit you had to connect with G-d's Creation through your pet. Thank G-d for His enormous chesed, through which you have been given not only food to eat and water to drink, but also a heart to love, and a world full of good and dear things, creatures, and people to love with it.
Then pay forward the chesed by doing a mitzvah.
I'm unaware of a specific composed prayer. However, in addition to the answers already suggested, I would recommend doing something in memory of your pet, that's done similarly, in memory of people; namely, tzedaka - charity.
Many people are in need of a service animal for various reasons - blind, deaf, psychological reasons, etc. Many cannot afford to purchase one, maintain them or care for the animal's medical condition. So, these service animals have a vital role in the human's health condition.
There are many charitable organizations that are devoted to helping these animals, and, in turn, these people, in various ways. Or, you can organize your own charity. It's a wonderful form of tzedaka!
The tzedaka, BTW, doesn't need to be monetary. Homebound people can't walk a service dog. You can donate your time to do this for them. The owner and the dog will both be extremely grateful.