1st Bike recommendation

HI, Daren, and welcome to BT.  Congrats to your daugther...mine is a sophomore and also retired (she didn't want to swim D1).

For a first bike I'd go new if possible, but I understand there's still some degree of supply chain issues on bikes in general so you might have to wait.  Used are fine for sure.

The most important thing is to get one that fits, the more triathlon-specific (i.e. with aerobars) the more that is important.  Height of your daughter is important but so are her proportions (leg length vs. torso vs. arm length) and flexibility (and if you're swimming daughter is anything like mine you don't need to worry about that).

My first triathlon was completed on a Gary Fisher hybrid that was $300 at a sporting goods store.  I then graduated to a road bike (a Fuji) that I rode for six years and then got a better road bike (a Ridley) before finally pulling the trigger on a triathlon bike last year.  I've been in the upper 1/3 of bike splits so to a degree it's not that important.

I'd take your daughter to a good local bike shop (LBS) to get a basic fitting at least, which will narrow down the bike brands, models, and sizes for you.  I'm a praying mantis with alligator arms so my choices were pretty weird (I'm 6'3" riding a size Medium bike with an extra long seat post).  From there you can choose whether she wants the bike for multiple uses (like commuting or just riding around), road riding, or very specific triathlon bikes.  They're more $$$ in that order generally for the same level of bike.

I'm sure some others will chime in, but don't break the bank until she's ready to really commit to the sport.  Of course, if you're independently wealthy then have at it. 

How nice a present are you looking/willing to give?

I got my Klein Aeolus used for $500 back in 2006. It’s a ’90’s vintage Dura-Ace setup. I liken it to a ‘63 Sting Ray. It was one of 2 out of 2000+ bikes at IMCdA in 2009. 

A buddy told me a few years later to “upgrade” but honestly the bike was never my limiter. 

you can get new tri-bikes for $1500-8000. Race wheels can be another $2000. 

In her case, she might end up on a sponsored bike and not ride this gift much. 

As the other poster said, fit first. There are some online calculators that will get to about the right size. 

I also started triathlon with a bike that didn't break the bank.  I got a used triathlon bike for $300.  I had done any cycling previously so I didn't know what to look for.  I know I could find a decent street bike used for under $300 and so I though that would work for me and that I could had clip on aero bars to use for triathlon.  When I started to look at aero bars they were $200 + so they were almost my full budget for a bike, so when I found a triathlon bike for $300 because I figured it would save me a lot of time and money in the long run. 

I took the bike to a local bike mechanic and he was pretty impressed with my find.  He said that the seat on that bike was really nice and that if I ever sold the bike that I should swap the seat out for a cheap and and keep that seat for by next bike.  So I had to look up the bike and the seat was $350 new.  The mechanic told me that the two things that he always said were wroth spending money on were a good seat and good wheels.  He said that they can always move from an old bike to a new bike so get the best that you can get.  

I have found that you can get a lot of "extras" with a used bike.  The bike I am riding now came with race wheels.  They would run $2000 new and I got the wheels and the bike for just $1,600.  You also sometimes get bike pedals, storage boxes on the bike, hydration systems, etc.  

My bike mechanic told me that there are lots of companies that make really good bikes.  He named about 10 of them off the top of his head and said to not try to shop by brand.  Yes, fit is the most important thing.  The way you know the fir is good is that the bike is fun to run.  If the fit is bad the bike is not fun to run.  The Local Bike Store can help a lot with the fit and make sure that what you get will be good for your daughter.