I could buy her an old Base Station on Ebay, I just need to know the oldest model number/nomenclature that will work with a PC running XP without a Mac present after setup.
All AirPort base stations (oldest-to-newest: Graphite, Dual Ethernet (Snow), & AirPort Extreme Base Station) are based on the 802.11 standard. The two oldest only support 802.11b, and the newest supports both 802.11b/g. So, depending on the wireless capability of the laptop, it shouldn't have any trouble connecting to any of these. (ref: http://docs.info.apple.com/article.html?artnum=107908)
1) If it will work with the HP laptop but no Mac there at her house.
Yes, but her laptop will require the AirPort for Windows software if you plan on doing any administration of the base station from that laptop.
2) Can I set the Airport up for her there with my Mac laptop, then come back home? Leaving only the HP there to access the internet wirelessly through the base station?
3) And bring my white 2001 iBook to work with some other wireless router I will buy (maybe an Airport if I can afford it), without loosing the settings for my mom's base station should I need to manage it at her house sometime in the future?
4) To re-ask #3 above, how would I manage and remain intact the setup/management information of two separate base stations at two different sites with one little ol' iBook?
Within the AirPort Admin Utility you can create multiple AirPort configurations and apply them when desired.