How old is the child? Have they been doing a budget for at least a few months? It's a lot easier to see where you can cut back when you know exactly where you're spending money.
A lot of books recommend trying to live on one person's income before the other person quits. I agree that you can defnitely do it if you can live on the spouse's income, but even if that doesn't work, it can still be possible. There are a lot of expenses related to working, besides the obvious ones, like daycare. If a woman comes home and makes being a SAHM and running her family on a budget her job, there are a lot places where it's possible to cut expenses. Of course, how much they'll be able to cut back depends on their life style right now.
One thing she can do is compare the percentage of her DH's income on different things and compare that to what financial experts recommend. You can find a tool from
Crown Financial Ministries that lets you enter your annual income and then see what dollar amount they recommend you spend on food, cars, house, etc. You can find similar recommendations from many financial advisors. Some families will have to decide to make some pretty dramatic lifestyle changes, like moving to a smaller house or driving one car, to have one person stay home with the kids. Some people are fine making that decision when they make it for a purpose and realize that they'd rather stay home with their children than live the way they used to, and others aren't. To me, it does make it easier to make that decision when I realize it wasn't something I was forced into but rather something we decided we wanted.
Like I mentioned before, depending on how much they're used to spending, it may be possible to really cut back once she has more time. She'll be able to cook from scratch and shop for bargains. One thing that really helps us is making a few meals ahead of time and keeping them in the freezer. When I'm really busy, like when I'm about to have a baby, I'll put enough in the freezer so that I won't have to cook for a few weeks. But normally, I just keep a few things in there so that when we end up being really busy and I can't possibly cook dinner, I can get something out of the freezer instead of ordering a pizza. I have stuffed pasta shells and homemade Hot Pockets in my freezer right now.
If she has a baby in diapers, I find that using cloth diapers helps me save a lot of money and time. I don't have to clip coupons and always drive to whichever store has diapers on sale. I do have to wash them, but that's just an extra load of laundry. I washed once every 3 days when I had one in diapers; now that I have two in diapers, I wash every 2 days. If you want more information about cloth diapers, let me know. I just PM'd a lot of information about them to another DISer a few days ago.
Like someone else mentioned, there are a lot of good books out there. I really like Miserly Moms. There are several good cookbooks that can help, too. I think that most (not all, but most) American families can afford to have one parent stay home if they really want to. Not every family wants to make the sacrifices necessary, and that's fine, but there are a lot of ways to get it done if it's a priority for you. I always counsel newlyweds to live off just the husband's salary from the start if they plan on having the woman stay home at some point because that does make it a lot easier, but it's still possible for people who didn't do that.