There's a few different answers you could try for this, ordered loosely by degree of genetic alteration needed. As far as I'm aware, none of these would be directly connected to the age at which puberty begins, although the later ideas might have odd side effects. A warning: I'm not a geneticist, so I can't tell you exactly what genes would have to be modified to do this.
Produce more viable eggs
This one is fairly simple. Female humans (I don't know if this is true for other mammals, but it probably is) are born with a finite number of ova, perhaps a few thousand. A lot of these are lost in the early years, until at puberty the typical woman has a few hundred viable eggs. At one a month, this will last for a couple of decades.
If you can tamper with the genetics a little, causing female babies to be born with more eggs and/or making a larger percentage of them turn out to be viable (which might call for larger ovaries, but don't take my word on that), you can extend the childbearing period, at least in theory. It won't cover an infinite number of years, but I'm not under the impression you're looking for outright immortality. This may also require tampering with the mechanism that causes menopause, since I'm pretty sure that running out of eggs is not the only factor.
Modify the ovaries to produce eggs continuously, much like sperm
This one is self-explanatory; if women keep making eggs, they won't run out. This is the only solution (out of my list) that would work on the scale of thousands of years; the others would eventually fail if a woman had a few hundred children (by which point many would probably have perished of childbirth-related complications, but that's not really relevant here). There are potential side effects (older men generally have lower-quality sperm, I think due to DNA replication errors mounting up over time), so keep that in mind. Again, you'll have to make sure menopause isn't triggered by other means.
Change the reproductive cycle
Now we're getting into serious genetic modifications; this might be outside the scope of changes you're willing to run with. There's a couple of different paths to go with this, but the basic idea is to cut down on wasted ova by reducing or eliminating cases where an ova is released without being fertilized.
Induced ovulation, such as what is seen in cats (basically, sex causes them to ovulate), is one possible solution. I hope it's obvious why this would result in a much higher percentage of released ova being fertilized and (barring miscarriage, stillbirth, etc.) resulting in children.
There's also the idea of making ova linger for more than a day or two after being released. This would require some notable biology changes, since the woman's body would have to remain capable of nurturing a fertilized embryo; you'd have to avoid menstruation for much longer.
Pushed to the logical extreme (the embryo lingers until fertilized), you'd have women who were either pregnant or fertile at all times after puberty, and who had no periods at all. There's a reason why this doesn't happen in nature: maintaining a fertile state in females requires lots of blood-rich tissues in the uterus and is very energy-expensive, to the point where shedding it regularly and regrowing it a few weeks later (periods) is significantly more efficient than simply maintaining it continuously. Women would end up eating a lot more.