Hungary has beautiful needlework. There are many different styles, but I would recommend looking for https://matyodesign.hu/ products. The company is located in the little village of Tard, which now has a UNESCO World Heritage designation.
You can find their products at the popular souvenir shop: Memories of Hungary and other places throughout the city.
You can find their Budapest locations here: https://matyodesign.hu/contact/?lang=en
Full disclosure: I know the owner and have been fortunate to attend one of their tours. I own several pieces of matyodesign products and they are all so special to me. You can even find them in the airport.
One of the locations listed for matyodesign is: https://folkartkezmuveshaz.hu/?lang=en
This is a fantastic shop with all kinds of Hungarian products.
Near the Synagog is a really great store: Holló Műhely (you can find them on Facebook)
On the Buda side, you can visit: https://prezentbudapest.hu/
For really special jewelry: https://dorivisy.com/
You will never go wrong gifting Hungarian paprika (I prefer sweet over smoked or hot).
For the chocolate lover in your life, be sure to find a https://stuhmer.hu/en shop.
Lastly, the Great Market Hall http://budapestmarkethall.com/great-market-hall-budapest will have everything you could possibly want.