The phrase is correct, just unusual. People use it to be unique, and to make mundane questions like "how are you today" become a little more interesting. And clearly it worked, since you asked this question.
Don't discourage your mom from saying this phrase. It's good that people use phrases like this, because it makes a question that should truly be asked wholeheartedly stand out more. I don't really give the question "how's your day going?" much thought. I just automatically say "fine". But a phrase like "how's your today?" turns on the thinking machine, and my answer is sincere. It's also a much more lighthearted approach to the daily question.
I know I did a lot more than answering your core question, with a lot of opinion-based stuff, but I actually think it's something one should consider. Using sprighty and unusual phrases instead of the grey, mundane and regular phrases we've become so accustomed to. Give it a try.