The Creative Woodburner from Walnut Hollow is what I started with and it's the one I use the most. It's very versatile, has temp control, and you can change the points very quickly. I also have the Versa Tool that was mentioned below; it's cheaper and comes with a case, which makes it a sweet starter tool.
Additionally, get her lots of patterns! (Amazon has many books on pyrography but "grownup" coloring books are fantastic resources, too.) Carbon paper is a must as well for easily transferring those designs onto the wood.
Right now, Walnut Hollow is also having a 40% off sale w/code WBCVIP40 in case you want to get her planks or rounds to burn on. They may or may not get here on time for V-Day so try Michaels or Amazon.
And because by the time she gets the goodies Feb won't be over, have her follow the hashtag #BurntFebruary on Instagram: It's a challenge where every other day pyrographers all over create/share their projects inspired by a particular word (Love, Animal, First Project, etc.). It's a great source for inspiration and ideas.
(Random but also along those lines: For more inspiration to gift her, look up the hashtag #BurntOctober [same concept as above but different prompts] and maybe try making a book with the projects from that hashtag like with Chatbooks, which I think may do it automatically.)
Not under $80, but the Colwood Detailer is an excellent burner. I've had and used one for years and love it.
There's the Walnut Hollow burner that meets your requirements, but I have no experience with it and can't comment on how good or bad it might be.
Other than that, there are a number of Chinese knockoffs on Amazon, but they can be hit or miss. I doubt that there is any quality control, and you can get an excellent machine or one that craps out in a few months or so.
This is the one I use. Colwwod woodburnerI've never really used the "heavy duty" side so I think it's more robust than I need, but it was a gift so I gladly use it! There are much less expensive ones by Walnut Hollow that I started with and they work great for the average user. If you burn a lot/verydark/every day, you may want something more robust than Walmut Hollow though. Hope this helps! Happy burning!
I think I found the middle ground. I've used this set up for over 2 years so can vouch for it's efficacy. The trick is that heating units are more or less the same it's to tips that matter. And the tips at colwood fit a cheaper heating unit.
Buy the walnut hollow creative versa tool Every now and again home Depot will have it for $50 and Michaels will match any price.
Buy the tips at https://woodburning.com/products/rt-hr-3-16?variant=12982428401733
They fit and heat just fine and the cheap walnut hole unit really does a good job of controlling the heat
These tips are inferior to wire tips - they're not quite as precise and don't heat up/cool down as quickly, but this is the tool that many people start with, and it will do an amazing job in the right hands. Once you've decided that this is a hobby you want to pursue, you can upgrade to a more expensive kit. I have the Colwood Detailer and love it, but it and other good burners are going to be way beyond your budget.
I just found this Walnut Hollow kit on Ebay, and if it's acailable where you live, I'd go for it.