Just adding my own two cents here but definitely anything is possible if you have the know-how.
However, since I'm looking at doing something not TOO dissimilar, I think PiHole is going to be the best option for you.
To answer a couple of things in the thread here already:
1) OpenDNS (or any alternative like this) isn't an option for your because with the default BT hardware you can't change your DNS settings at the router level.
@Caged - the changing IP isn't too much of an issue as you can get a OpenDNS client that updates the IP address as it changes. This does require the client to be running for this change to take effect but at most he'd likely only lose one or two days data if it was a commonly used computer.
2) Wireshark. If you don't know the difference between Putty and SSH then you're possibly starting just a little low level to be playing with Wireshark. Not to say it isn't possible, but I'd say this is the more technically advanced option with no other benefit.
The Raspberry Pi with Pi-Hole is a pretty neat solution imo. It will let you see who's accessing what without being too deep (in a sense of literally seeing everything she types online - which Wireshark could almost let you do)
As a side note, while you do get web-based access to Pi-Hole you can also install "TeamViewer like" software on your RaspPi to allow you to access the desktop interface remotely. But, as others have suggested, your definitely going to need an HDMI capable monitor in the first instance to get it up and running.
Search the Rainforest for "Raspberry Pi 3 Value Starter Kit" and pick up everything you need for £43.
Follow the advice from
@BigT (disable DHCP on BT router etc) and in addition to seeing what's being accessed, you'll have a nice ad-free home network.
However, consider how "open" you want to be about what you're doing. The fact that every user in the house will get the odd "access blocked" page when trying to view certain things online consider that it may change their browsing habits (if they think they're being watched)
You might want to tell a white lie that you're adding this device to improve your own internet browsing experience OR tune down filtering (maybe even turn it off) so nothing is apparent (TM RoyMi6Corp - "Lying to children for 30 years")