I've mostly used booking.com for my two recent trips to Europe. You can choose Central Barcelona as your area and filter by price and qualities that are important to you.
I stayed at the Hotel Catalunya on C. Santa Anna this August. C. Santa Anna is about one block off Pl. Catalunya. It's a short but busy pedestrian street with shops and restaurants, but I heard no exterior noise at all because my room faced a sort of air shaft (I had that a lot in my budget hotel rooms). The hotel is extremely close to the Aerobus stop and the tourist office under Pl. Catalunya, and the Barri Gotic and Casa Batllo are both nearby (though in opposite directions). There's a two-floor supermarket on the Ramblas that's also not far away, and restaurants all over the place.
It's a two-star hotel, but in my experience two-star hotels in Spain are generally fine if you're looking for something functional rather than fancy. I had a single room and it was extraordinarily tiny (you can see what a big price difference there is between single and double rooms right now), so you'd definitely want to book a double. The elevator worked, the Wi-Fi worked, the a/c worked (not that you'll need it). The shower may have been small, but I'm so used to that that I just don't notice anymore.
An additional issue is that the hotel may try to charge you in dollars rather than euros, and you do not want that. So if you decide to try this place, address the issue up-front, via the Comments field when reserving and when you arrive at the hotel. Then check out on your last night if you have an early departure the next day so you have time to take a close look at the receipt when it is presented to you.
Careful: Booking.com is offering only non-refundable rooms at the hotel right now for the days I chose. Check the hotel's website to see whether you can do better. There are a lot of other pins on the map around Pl. Catalunya, so you do have some options.
I suspect that the length of your stay may be affecting room availability. It's something that concerns me when I'm expecting to stay more than 4 or 5 nights in one place.
Edited to add: It's a bit difficult to find the hotel's website, but eventually I succeeded.