It's great that you are thinking of her. A card can be a nice gesture. Do you have any really nice pictures of her and her dog together? If you do, how about getting one made into a nice print and framed as a Christmas present? If you have a few, you could make a little scrap book. She might also appreciate a donation to a suitable rescue in memory of her dog. Depending on what she decides, she may want to scatter her dog's ashes in her garden, bury them or scatter them somewhere special (such as a favourite beach or park). Your aunt might appreciate your company and support when she is ready to do this.
Really, the most important thing is what you are already doing - thinking of her and caring about her.
Don't rush to suggest she gets another dog, just let the grieving process happen, and let her decide when/whether to talk about it.