Once you've sorted out your Tv and Freeview box all you need then is a good aerial to connect to it.
I managed for a while with a battery powered indoor aerial for a tenner off ebay but didn't get all the channels.
It depends ho strong your signal is .I'm just mentioning this as a stop gap solution.
In the end we had to get a feed from the main roof aerial into two upstairs bedrooms.
If you just want one feed you should get away with just getting an aerial wire run down and into your room.
(They run it down the roof the drill a hole through the outside wall into your room).
As we wanted two rooms as well as the downstairs TV from one aerial we had to get a signal booster as well.
So the firm we called out went up on the roof ,ran a wire down either side into the two bedrooms and put a signal booster in as well which is just a little box sitting by your telly.
All this cost £90 in total just to give you an idea.
If you only need one extra TV as it sounds you should be able to just get a wire from the roof into your room.The roof aerial should have a 'splitter' to plug an extra cable into.
If it's getting pricey,why not look on Freecycle for a TV ?
It may not be the fanciest but it'll be free and there's usually a good few on my local one.
Phone a few places and get some quotes so when you approach your mum you'll know what you're talking about!