Jay Huggins' answer makes a lot of good points. I like to add something that might be important to you as a parent to know.
Having followed lessons for around 16 years I'm not ashamed to say that the first part of learning drums is the most boring part. That is, if you do it the way you should: basic rhythms, rudiments, the like. Only after you've mastered those basics you can move onto standards. There's too many people who try to play a song, before even being able to play straight eights in tempo. So a good approach would be to get started with a pad. Doesn't even have to be electronic!
I used to take a Play Doh like substance with me everywhere, roll it out in a circle of around 6" (15 cm), and you have your pad! (Note that Play Doh might be too soft and would get holes in it too quickly. Unfortunately I don't know the brand of the actual substance I used any more.) This is great for rudiments, and basic rhythm practice. Most kids have a smartphone these days, so if she could install a metronome app on her phone, get some ear buds and the Play Doh like stuff and you're set for less than $25 (£16/€25)! This gives you a head start to start saving for a bigger set (acoustic or electronic).
Another approach would be, as you said, to get an electronic pad. In one of the rooms at my school we used to have them, a "snare", hooked onto a "bass drum" and a "ride cymbal" (in quotes, because they were electronic). It also had a built-in metronome, and audio-out for headphones or ear buds.
Whichever approach you choose, as Jay Huggins said it is a great approach - at the start of her lessons. She might not like it, and it gets boring quickly, but I guess it's something you have to get through when learning any new instrument. When she's getting better, and she really uses the equipment you got her, you can start thinking about saving for better equipment. For the first two to three years for instance, she should be able to manage perfectly without a whole kit. If she gets bored, or wants to practice on a real set, maybe you can arrange with the tutor a free moment for her, e.g. once a month that she can do whatever she pleases on the acoustic kit at practice.