Buying souvenirs or gifts to send to your family back home can be tough - Brasilia is really a government city, not a tourist city. But there are options if you know where to look.
1. TV Tower
Address: Feira de Artesanato da Torre de TV SDC, Eixo Monumental, Brasília - DF, 70070-350

Image by Daderot from https://en.wikipedia.org/wiki/Brasilia_TV_Tower, used under the Creative Commons CC0 1.0 Universal Public Domain Dedication.
Down the hill from the TV is the TV Tower Market. There are many little shops selling all the typical tourist tchotchkes, such as keychains, pencils, t-shirts, etc. Keep an eye out for the handwoven baskets - those are made from a grass that is native to the cerrado (Brazilian high plain).
2. Fundacao Athos Bulcao
Address: Asa Sul Comércio Local Sul 404 Loja 1 - Asa Sul, Brasília - DF, 70297-400
Website: https://www.fundathos.org.br/athos-bulcao
While you're walking around the city parks or at the major city historical sites (such as the churches) you may notice a distinctive blue tile pattern. All those tiles were designed by Athos Bulcao, an artist from Rio who was commissioned to design these tiles for Brasilia. The Foundation's shop has not only some of the tiles themselves, but also mugs and other gifts featuring his distinctive patterns.

The mesmerizing repetition of the Athos Bulcao tiles at Igrejinha Nossa Senhora De Fatima Church. Photo by Heather Torrance.
3. Verdurao t-shirt shop at Infinu art collective
CRS 506 bloco A loja 67 - just off the W3
https://verduraocamisetas.com.br/

Photo by Heather Torrance.
Verdurao has Brasilia-themed t-shirts with cheeky sayings and drawings, such as the iconic "I Heart NYC"-style shirt, only with a pink ipê tree in place of the heart.
It's on the street side of Infinu, an art collective selling high-end craft and home decorative items. There's also a nice little coffee shop and a gelato place, and a tattoo parlor inside.

4. Endossa Collaborative Store
Endossa currently has two locations in Brasilia, one in Asa Norte and one in Asa Sul. The items they carry vary depending on which artist or producer is renting the booths at the moment, but all of the products are high-quality and unique.
SCLN 310 Bloco C Lojas 14-20
SCLS 306 Bloco A Loja 30
Please leave other suggestions you may have found in the comments!