Slippery elm, quinine, coca, goldenseal, American ginseng, cohosh, echinacea, witch-hazel, much more. A nice essay on Native American anti-inflammatories is "Inflammation and Native American medicine: the role of botanicals". Jack Weatherford's Indian Givers: How the Indians of the Americas Transformed the World is a good overview.
One thing to keep in mind from the start is that "medicines" in the pre-modern era were substantially different from what we would classify as medications now. The modern conception is one of a pure substance that directly has an effect on the pathogenic agent or which alleviates symptoms of a diseased state. That latter definition is a good way to conceptualize the use of medicine in the past. Absent the idea of germ theory, the approach was to have an established pharmacopoeia which could be used singularly or in conjunction to combat both symptoms and recognized diseases.
In addition, the conceptualization of disease in the Western canon as stemming from disruption of humors or as a result of miasma meant this pharmacopoeia, however, could be secondary to other approaches such as bleeding or other non pharmacological approaches. Wear notes that, with plague:
... many medical and surgical procedures such as purging, vomiting, sweating, blistering, cupping and bleeding were designed to expel noxious humors or a plethora of humors from the body.
In other words, there was not an equivalent to modern pharmacology texts in 16th Century Europe. There were certainly texts which laid out medicinal uses of certain herbs and roots, even the appropriate times to gather them with regards to astrology, seasons, and lunar cycles. We do not have the kind of relatively simple algorithms of modern medicine wherein a bacterial infection indicates use of a specific beta-lactam antibiotic. Instead, we have conglomerate applications of vegetable and animal matter used in conjunction with surgical techniques, diet, and primitive epidemiology in order to effect changes based upon the understanding that certain environments produced imbalances within the body and those imbalances produced disease.
To reference Read again, this is why we have Lady Midmay touting a balm composed of "24 different kinds of roots, 73 herbs, 14 seeds, 12 flowers, 10 spices, 25 gums, 6 purgatives, and five cordials." Or why the equivalent of public health officials at the time recommended clearing waste from the streets, setting bonfires, and walking around sucking "cloves, angelica root, citron pills, or garlic." These were not pharmacologic remedies, they were miasmatic ones.
All of this is to say that we cannot and should not expect our modern understanding of "medicine" to track with that of that past.
That said, there are some notable examples!
One such example of an American remedy being picked up in a big way by Europeans is guaicum, a genus of Caribbean trees which were touted as a syphilis cure into the 19th Century. While the actual use of guaicum as a medicine for this particular application is a bit murky, it was recorded (along with sarsaparilla) as an indigenous cure by Francisco Hernandez, the Spanish court physician who undertook a massive project of categorizing and interpreting native herbology into European terms. Guaicum, however, has absolutely no efficacy against syphilis, unlike our next subject.
Cinchona, better known as the source for quinine, had been used in the Andes to combat headache and shivering before being adapted by Europeans as a remedy (not a cure) for malaria in various fashions. Eventually, of course, the active ingredient would be isolated and quinine-based drugs now form the primary prophylactic and therapeutic remedies for malaria.
One interesting case is of chocolate, which was not really used medicinally in Mesoamerica. It's exotic flavor attracted it to European medical practitioners, professional and amateur alike. The aforementioned Hernandez categorized it, in the European style, as "cool and humid," although that latter categorization would later be dropped. Coe & Coe, for example, recount a recipe by a mid-17th century Spaniard consisting of:
This beverage was part medicinal, part enjoyable; think of it the way people drink smoothies today.
The use of cacao as a therapeutic agent was indicative of the way prominent American plants were being adopted into the European consciousness. Cacao preparations paralleled the use of tobacco, as well as coffee, as substances introduced to Europe that were both medicinal and recreational.
Tobacco is actually another interesting case. While widely used in religious rituals, Ortiz de Montellano notes that it was widely used in Mesoamerica for the treatment of wounds, particularly of venomous bites. The wound would be packed or rubbed with ground tobacco as a curative measure. While this particular use does not seem to have translated perfectly into Europe, we can see the use of tobacco for a broad variety of other therapeutic uses.