A paper trimmer
A rub-on tool is nice, but not necessary
little X xyron (maybe not technically a tool)
eyelet setter (I have the 'old style' you use with a hammer)
anywhere hole punch
Loving my spring punch too
I love my cutterbee scissors, Fiskars fingertip control craft knife (but not the swivel one...I can't get the hang of it), tweezers, Heidi Swapp edge distresser, creative memories journaling pens, KI Gloo, glue dots, paper piercer, and punches. For some added fun that doesn't take up a ton of space, you could add cats eye chalk inks, some brads, and stickles or a stardust pen.
Is she just starting out? The other posters have listed many great basics. How about various adhesives? Photo splits, Zip Dry, pop dots, diamond glaze, glue dots, etc. She'll get to try out various things and maybe find a new favorite and they won't go to waste. I just got an invisible ink pen that I think I'll really like, so she might, too. Border punches are a great idea, too.
Use a ruler and draw lines, then journal and after a while the ink disappears. You can also use a spiral or wavy template to draw the lines for a different look for journalling. Most people just erase pencil lines, but I don't like the way it looks when I do it.
Wow, what a great question and what awesome answers. The only thing else that came to my mind (and I may have missed it) would be a good light source near where she will scrap. Nothing expensive but good lighting helps more than some think.
I really like my cutting mats from the dollar store. I can cut things and glue them on the same workspace. They roll up and are very portable. I use a glue stick a lot and the cutting mats just rinse clean.