Pretty much what they said, though the Remarkables is also a good contender -- if it's had enough snow by then, and assuming you're staying in Queenstown. At least the NZ school holidays aren't until a bit later (July 5-20), though the Aussie holiday crowds might be a pain.
There've been a few developments at the Remarkables this past summer which possibly make it a better bet for you and your mother, although it is set to open a couple of weeks later (June 20) than Coronet Peak. I'm generally reluctant to ski there unless it has a base of at least 85cm, whereas 25cm coverage at tussocky Coronet Peak isn't going to do nearly as much damage to your skis. Terrain at Remarks is a mixed bag and it doesn't have quite the range of continuous fall-line skiing of Coronet Peak and Cardrona unless you're hiking/traversing to them, but it is very well set up for beginners, and now also to intermediate skiers with the new Curvey Basin chairlift accessing what used to be some very nice hike-to slopes overlooking the base area below.
It's a longish drive up, but the road's just been partially sealed from the bottom (with plans to seal the whole thing in the next year or two, I believe), which will hopefully erase some of the more "interesting" features (corrugations) of getting there. The Remarks faces north, so it gets more sun than Coronet Peak, but the extra elevation it has over CP tends to keep the snow longer than you'd think. And the views are fabulous.
If the snow's still a bit thin, the snowmaking at Coronet Peak and Cardrona would make them the better bets. CP can get crowded (especially in the holidays), though the new amenities and cheaper passes at the Remarkables will probably draw a few people away. If you've got a few days there I'd visit all three -- they're all fun on a good day. If you don't mind the drive, I wouldn't necessarily discount Treble Cone either: people do in fact learn to ski there too, and from memory there's a big cruisy run off the main chairlift out of the base area which is ideal for intermediate skiers. Fabulous views there too.
Driving times from Queenstown are roughly 20-30 minutes (depending on weather, road conditions and traffic, and where in Queenstown you're staying) to Coronet Peak, 35-45 minutes to Remarkables, an hour or so to Cardrona, and an hour and a half or more to Treble Cone. If you're in Arrowtown, it's about the same to Coronet Peak, but a bit longer (5-10 mins) to the Remakables, and a bit less to Cardrona and TC. Wanaka is about an hour to Queenstown. Wanaka to Cardrona is about 35-40 mins (from hazy memory), and TC about the same or a bit more. (I'm open to correction on that last bit of info.)