The original thread was closed by the Webmaster due to a period of inactivity. Soooooo, do reference back to it for a lot more information. https://community.ricksteves.com/travel-forum/hungary/rs-forum-favorite-hotels-and-their-cost
Some updates:
The 4-star Maison45 is a new addition in a nice location. https://www.maison45.com/ Recently opened I like the area. Refundable rooms with breakfast start between $200 and $250 so not the cheapest, but looks to be a very nice hotel. Actually the suties starting about $325 are priced pretty well for what you get. There are only 4 standard rooms and 13 suites. I havent been in it (yet) but what it looks to be is the street facing side of a an apartment block is where the suites are and the half attic is the cheap rooms. If I am correct, maybe you get to share a bit of culture with the neighbors in the building ... but not sure.
Another that seems to be popular and is in a good location is the 4-star Three Corners Avenue Hotel. https://threecorners.com/budapest/avenue-hotel/ Starting at about $130 for refundable and breakfast.