I am sorry to hear that your mom's end is near. Pets do grieve, as others have noted. However, how you handle the situation depends a lot on the age of the animal and how long he has been with your mom. I do think that you should try to find a good home with an older person (though preferably a healthy one.) If that does not work out and the dog is still unhappy, you might want to consider putting him down. Do try to find a home for him first, though.
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"But that was yesterday, and I was a different person then."
Lewis Carroll, Alice in Wonderland