Your daughter is fortunate that she has such supportive parents. You will surely have a lot of nachas from her as she blossoms into an inspired Jewess and eventually establishes her own home.
While it is true that sometimes “newbies” to Yiddishkeit need some guidance to make sure they don’t go off the deep end, in this case there is nothing “ultra” about your daughter’s request. Every door in a Jewish home deserves a holy mezuzah!
Unfortunately, you are correct to note that years back it was common for people to have only one mezuzah on their front doors. We must remember that the opportunities for Jewish education in our parents’ generation were limited, and sometimes the traditions that they followed were uninformed. Thankfully, the Jewish education that you have provided for your daughter has inspired her to want to follow authentic halachic rulings.
We must also keep in mind that our parents were often struggling to make ends meet and may not have afforded the Mitzvot the attention they deserved. Admittedly, fitting out a whole home with kosher mezuzot might prove to be a pricey exercise, but the blessings that they will surely bring are priceless!
The responsibility to affix mezuzot on the doorways of the house rests on you as the owner. Your daughter can continue to live in the home and use all its rooms even if they do not have mezuzot. There is one exception: Since your daughter has been allotted her own private room, she would be obligated to purchase a mezuzah for its doorways.
In closing, I would encourage you to put up the mezuzot, not for your daughter’s sake, but for your own!
 Shulchan Aruch Y.D. 286:16-17. Of course, the doorway and room must meet the halachic criteria discussed in this book.
 See Rema Y.D. 287:2 and Shach 286:9.
 Teshuvos V’Hanhagos 2:537; Agur B’ohalecha 2:55.
 Agur B’ohalecha 30:53:124.