I'm going to leave aside the wisdom of doing this, and focus on the practical aspects.
Has anyone built anything like this?
An Android phone hooked up to Google Latitude would meet most of your requirements - small, accessible anywhere you've got an internet connection, accessible on mobile phones, phone can be turned off, phone is linux based.
Is there an open source solution?
There's OpenGTS, an Open GPS Tracking System. However, it's not obvious from their website what trackers it works with.
How would I go about building my own?
People have home-made basic versions. GPS modules can be purchased which give a reasonably accurate location once per second, or on demand, over a serial, usb, or bluetooth link; many mobile phones have gps modules already built in. Most tracking systems communicate over the cell phone system, either by SMS or mobile data connections. Of course, many mobile data connections are firewalled/NATed, so the benefit of SMS is you can transmit a query to the tracker. The disadvantage is the per-message cost, especially if you want regular location updates, and that it's easier to program the PC end of a mobile data connection. Cell companies also offer "machine to machine" data plans, but it's unlikely they'll want to deal with you if you're making a one-off homebrew system.
You could get a separate cell phone and GPS and make a homebrew device, like the one linked above, but you're unlikely to get things much more compact than buying a mobile phone with both built in.
If you're a programmer, my suggestion would be a mobile phone running Android, and using the GPS APIs to read the location and send it off to your server.
How does a tinfoil hat wearer engineer such a device to make sure Big-Brother isn't watching too?
Pretty much every mobile tracking system uses the cell phone network for connectivity, because it's more widely available than WiFi, and more affordable than a satellite connection. If you're paranoid about privacy, you should be worried about cell phone triangulation, as that would be the most practical way for "big brother" to track people; so to be paranoid, you can't use a cell phone connection, which will make your design task substantially more difficult. It would be far easier to get a mobile phone, set it up with Google Latitude, turn it off, and tell your daughter to turn it on if she gets lost.