It's hard to answer without knowing whether your current job:
1. Is a career-track job, and you love it, and leaving it now would damage any chance you have of moving up at this job or continuing a long-term career you've worked and trained for;
2. Is a job that isn't part of a career for you, or something you love doing, just a job mostly for some useful extra money;
3. Is a job that is vital to make ends meet in your household, and without it your family will have a tough time.
4. Is a job that isn't big-time for money or career but it's something you like doing and it keeps you sane, because you know that staying home full-time just isn't for you (and that's fine!).
Find out the answer to those thiings and that will help you decide. Since you say that it doesn't make sense financially to use day care, it sounds (sorry if I'm wrong) as if the job doesn't pay a whole lot, if what you mean is that your paycheck would be entirely eaten by day care costs. Is that the case? If so, and it's not economically vital to the family that you work right now, you could indeed consider quitting. Yet you say you'd need to work nights and weekends, so maybe the money is vital.
But be aware of this -- after the hip replacement, she will need quite some time to recover and frankly may not be up to looking after an active toddler even when she has gotten the surgery and recovered. So you might be seeking day care anyway even when she's over the operation. Think through her age, her general health, her plans for the rest of her life (does she foresee watching your child as your child gets older? Once he or she is in school? Does she want to stop watching your child at certain point? Was this always intended to be a temporary thing that's turned more permanent than expected? And so on.) Be really frank with yourself and with her about whether she can or will be able to keep caring for your child.
As for the operation -- the sooner the better. I've had numerous relatives go through hip replacements and the longer she waits the worse her hip will be and the more difficult the operation will be. The recovery will take longer if she waits longer to get the operation. Sounds like it might be time to take your mom out of the child care equation for good. She might insist that she will be up to it, "Don't take my grandbaby away," etc., but do look realistically at overall health and the years ahead. She might actually be relieved to be let out of daily care duties though she may resist at first.
But only you can answer the questions about how vital your current job is and whether you really must work nights and weekends. If you work nights and weekends, will your husband care for your child? When will you see your family if you're working nights and weekends? Is there any way that you can not work at all? There is a lot to weigh but it sounds like taking your mom out of the equation might be necessary at some point anyway. Good luck.