For what I'd consider "must have" books for almost any campaign, you're most of the way there.
Street Magic and
Arsenal open up most of the options characters are likely to want. If you pick up
Augmentation you'll have the other of the "big three," in my opinion, offering up a plethora of goodies if your player is interested in some artificial muscle or some outright chrome.
Unwired helps out with the Matrix-sorts, but I'm not (personally) entirely convinced it's the same level of gotta-have-it book (perhaps because no one in my group's really been eagerly reading through it, despite having access to it). Really, that'll depend on what sort of character Gecko's making.
But really, after those basic expansion books? It's going to come down to what sort of campaign you've got, with what sort of character. If you're interested in espionage and the subtler side of the shadows,
Spy Games will be right up your alley. More into the in-your-face street side of things, with a lot more flavor from the Sixth World's tv shows, musicians, etc?
Attitude. Thinking about a run to Vegas, or staying on the cutting edge of the Horizon metaplot?
Twilight Horizon. Interested in London, or having her tangle with shadowy groups with secret agendas (instead of your usual clean-cut corporate Mr. Johnson)?
Conspiracy Theories might be what you're after. As a long-time fan of the game, maybe you're interested in
Street Legends to set the stage with some of the old fiction characters and other shadowrunning elite, to let her new character rub elbows with some of the more (in)famous NPCs out there. Is she making an adept? She'll love you forever if you help her out with
Way of the Adept (as someone already mentioned, which I appreciate).
Etc, etc. Once you get past the "second-tier core" of
Augmentation,
Street Magic, and
Arsenal, it all comes down to what specifically you're interested in focusing on, in my opinion.
Augmentation, Runner's Companion, Unwired, War!, Attitude, Spy Games.... Heck, just get them all, that way your players can actually have an enjoyable time. Limitations are only fun for the GM.
Not all of those are going to be applicable for every campaign, though, and -- more importantly -- it's not up to the GM to have to buy them all so that their players "can actually have an enjoyable time." He's mostly just running a game for his gal, here, it's doubtful he's going to
need every page of every book ever released in order to keep her from being miserable throughout the campaign.
