okay. i'd start slowly. it's a girl, so her heat requirements are probably higher than what the boys are used to. also, you ought to break it down into different areas of the body, and think in onion layers so that you can adjust for the weather conditions easily.
-> gloves: for gloves, i got my gf hestra gloves. i can't stand them as they are way to hot for me. she considers them 'warm enough'. you ought to consider getting a thin inner glove (5 fingers) that is fleece, neopren or similar, plus an outer mitten glove that is water and wind-tight. the outer one should be long enough to easily extend over the sleeves of the jacket.
-> hat: get some that notably are longer/further down over the ears. i like woolish ones on the top, with a windstopper fleece around the ear area. these works also good for my gf, but she has rather short hair. if your daughter has lots of hair ... don't ask me. it's usually a pain in the neck to control that mess of hair in strong wind + winter
-> jacket: again, multilayer. at the mountain rescue we get issued a rather thin inner wind-breaker fleece that you can easily wear while skiing in better weather. just need reasonable clothes underneath and males usually are fine with that. for worse weather, you should get something that is (like the outer mittens) wind and rain proof. also, most of the more recent jackets have zippers under the armpits so that you can adjust the climate a bit here ... so you can be save against rain but yet have fresh air towards the body. personally I highly recommend you to make sure that both the inner shell as well as the outer storm-jacket notably go below the buttocks. i hate a cold back, and even worse cold kidneys. having the jackets extend over the buttocks helps alot. especially when you sit on the ski-lift, short jackets tend to not properly cover the lower back. with respect to size: rather buy them a bit too large. if it's cold, you can then easily add a thick pullover and be fine if your daughter needs more warmth. a pull-string at the bottom of the jackets (almost all quality jackets have something of that sort) allow you to have a little bit larger jackets without a disadvantage (say - wind blowing towards your torso)
-> trousers: i prefer trousers that: a) are wide enough at the bottom to cover the ski boots. that way you make sure that no snow can fall into the boots from the top. here, a zipper or some straps usually do the trick. starting at the knees towards the bottom, it should be a decent material such that a stumble with the skis does not necessarily cut up the trousers on the very first day. it's also a bit of a safety measure for the legs. from the knees upwards, I prefer a rather thin material that allows the air to get out. skiing is sport and my legs tend to become very warm. so while i keep my torso warm, i like legs to be closer to the fresh air to get rid of some of the heat. get a pair of 'over-trousers' that can be opened on the sides entirely and allow you/her to be entered/added without getting out of the boots. they come in little bags, the volume about 1l. if the weather turns bad, get them out, get into them directly on the slope, and no matter whether it's rain or a stiffer wind ... you'll be sit in it. those over-trousers ought to be strengthened at the bottom on the inside of the legs to withstand friction from extended rubbing of the ski boots.
-> underwear: on the top, get her a full-sleeve merino underwear shirt. for underneath the pullover (or whatever she wears as top layer). for the undertrousers, get 3/4 trousers. within the ski-boots she'll wear thick socks and the boots go up rather high. 3/4 undertrousers are short enough to not create a very thick (or rather too thick) amount of clothes at the area where the boots end. long ones usually make everything ... too filled up and reduce the joy of skiing. shorter ones are hardly worth it. make sure that those two pieces (shirt+trousers) have a rather looseish fit. they ought to trap air but should _not_ restrict blood flow to the extremities. this actually applies to all clothes. there very moment blood flow is restricted, she will 'feel' cold. so again: rather a loose fit, but make sure that the air is trapped.
final comment in that section: during the last ski tours (and summer climbs) with my gf, i noticed that her sports bra was not drying fast enough. especially during extended amounts of hikes, it became ... unnecessarily unpleasant for her to wear that damp piece of cloth. i checked back with my female climbing +hiking friends and, interestingly, several of them all agree/recommend that 'anita' is a good company to buy sports bras that dry really quickly and provide a very good hold of the chest area. not sure whether your daughter needs that, but it's something you might want to keep in mind.
so while i didn't talk about any company names (almost) - especially because i don't know to which brands you have _hands on_ access to (so that she can try the equipment in person) - i do hope that i gave you some points to consider.
maybe my final suggestion: make sure that _she_ enjoys it. and not that she does something that only you enjoy. that's why you ought to include her in the decision make for the equipment, colours etc. explain/show her why you'd recommend this and that, but still let her _take part in every aspect_. if she does it on her own, she might do it _with_ you. and not only _for_ you ...
