Well, that's easy to answer when we realize that "First Quarter" isn't talking about a 'quarter' of the Moon being illuminated. It's talking about a 'quarter' of the time that the complete cycle of phases takes.
"First quarter" means that 1/4 of the time from one New Moon to the next one has passed, and there are 3/4 of the cycle left until the next New Moon.
-- roughly 3/4 of a month to go
-- roughly 3 weeks to go
-- actually 3/4 of (29.531 days)
-- actually 22 days 3.6 hours until the instant of the next New Moon.
But Haley doesn't actually need to wait that long. You can't see the moon in the sky for 1-2 days before the New Moon and 1-2 days after it. So Haley really only has to wait like 20 days until there's no moon in the sky to interfere with her observations.
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Did you notice what I noticed ?
What if Haley stays interested in Astronomy for many years, gets really serious about doing observations regularly, and one day she discovers a comet ! These days, comets are named for their discoverers, so if that happened, we would have TWO of um !