A few thoughts:
I have seen more than one novice put off cycling forever by a crap groupset. Figuring out how to find the right gear out of 20-odd is tricky enough without the bloody thing constantly mis-shifting. The situation would be somewhat ameliorated by your presence, but some very low-end bikes just cannot be persuaded to shift reliably. You can find hub-geared bikes for £300ish, which can make a massive difference for some riders - it's a simple system with a single shifter and you can change gear while stationary.
If she is genuinely fit, she might outgrow a cheap hybrid fairly quickly. A fit non-cyclist can very quickly turn into a fit cyclist, on the proviso that the ramp-up is gradual enough to allow the contact points to toughen up. I know runners who have switched to Duathlon, gone straight onto a full-on TT bike and become very quick cyclists within a matter of weeks; Muscular adaptation happens quite quickly if the underlying cardiovascular fitness is there.
Novice women tend to specify a step-through frame because they have the erroneous idea that a proper diamond frame is somehow difficult to mount and dismount, often because of memories of learning to ride a bike that was much too large and had no standover height. That belief can be quickly dispelled, simply by teaching the correct way to mount and dismount. A fit rider is likely to become frustrated with a step-through, because of the extra weight and lack of stiffness.
I'd suggest taking a few trips to different bike shops, to allow her to take a look at different types of bicycles and get a feel for things, and to allow you to check out the quality of the components. If she feels confident in doing so, a few test rides would be a really good idea. Novice cyclists tend to be very bad at predicting what kind of bike they'll actually enjoy riding - a lot of people who think they want a nice gentle sit-up-and-beg get fed up with the weight and end up on a racing bike; Conversely, a lot of people buy a racer and end up wanting something more relaxed and comfortable. There's no substitute for getting out there and handling a lot of different bikes in the metal.
As regards specific brands, I think Decathlon offer the best value by some margin. Their leisure and city bikes are very well specced for the price and tend to come with lots of very useful extras - mudguards, chaincases, pannier racks, kickstands, even dynamo lighting on many models. Their fast hybrids and road bikes are astonishing value, if you go down that route.
If she is at all apprehensive about going out on the roads, training is a really good idea. It can be a massive confidence boost to know that you've learned the right skills from a professional instructor. Many local authorities offer free or subsidised training.