How to decide when to put my Mom's dog down?

Yes, it will effect your Mother and she will probably decline because of it. Consult a veterinarian who can tell you how much pain the dog is in. Some canine cancer pain can be controlled with pain meds, so it may not be hopeless. If it must be done then grieve with your Mom. Let her grieve as long as she needs to. Losing a beloved pet is like losing a child. Non-pet people seldom understand that. I don't think a puppy would make a difference except in more work for you. Wait until your Mom asks for another dog and then consider a rescue, rather than a puppy.

Sad challenges. I'm someone who has found love with my pets easier than with people, and I have major issues with maintaining connection and care, through sickness and in health. You don't say if she lives with you, or you with her - but one thing I can think of is, expect time to talk with your mom about how the dog is suffering, if you see that. Dogs can manage with tumors and cancer, and can receive rest and reassurance from being close and cared for. It's not easy to tell, unless you actually see wincing from pain. But I would start conversations about the situation with your mom - maybe over a couple of months of repeated conversations, she will have on some level taken in the idea that the dog is suffering, and that her job as owner is to be noble, and take care of the decision to put the dog down. If this dog has been close to your mother, I would not make a decision separately - but just keep bringing up the illness and pain and care, even people with alzheimers have some times of connection to life, and mothers often understand death better than daughters, they have faced it many more times.

Agree with vikilynn - if Mom is attached to dog it could make AD worse.
As long as dog is eating and "bodily functions" are working - other issues especially so long after diagnosis don't matter so much.
If dog goes, and if you can handle taking care of another - please consider adopting an older dog (rescues, and animal shelter have lots and lots)
A dog offers non-judgmental support and love. If you can help to take care of it, it may keep Mom happier. Not to mention helping to save the life of an "unwanted" dog.

I wouldn't do it unless the dog is suffering. It can be traumatic for ur mom and cause further. Decline. It works both ways too. My father died a year ago and his sidekick couldn't handle grief and died days later

Are you the one that is caring for the dog? You can only do so much. Don't beat yourself up. Mom might enjoy a new puppy, if you can care for it.

The only time I would put a pet down is when the pet stops eating.... that is how a pet that has diagnosed medical problems tells you it is time to move on.

Otherwise, if the pet is happy and doing fine, even with the tumors and hip dysphasia, the dog could enjoy another year or two, and your Mom could continue to enjoy her pet.