Low cost hotels/hostels in Barcelona for young women in July?

My daughter and her friends (18 at the time) stayed at the Hotel Nouvel. At her recommendation we stayed there ourselves earlier this year, and it's lovely. In the Gothic Quarter, very central.

Our daughter traveled around Spain and loved it - we didn't have any safety concerns, nor did she have aby problems.

Might be worth posing your query on the Lonely Planet website as well.
https://www.lonelyplanet.com/articles/staying-in-hostels

... you're extremely late in the year for this. Everything has probably been booked for ages by now.

Barcelona has a huge hostel scene, and some places have become almost legendary among backpackers and young travellers. Famous names include Kabul Party Hostel in Plaça Reial, known for its party atmosphere; St Christopher's Inn Barcelona near Plaça Catalunya, hugely popular with international visitors; and Generator Barcelona, a more modern “design hostel” style option. Other well-known hostels include Sant Jordi Hostels Rock Palace, Yeah Hostel and Onefam Ramblas, all of which are especially popular among solo travellers, Erasmus students and people looking to meet others while visiting the city.

In any case, if you're hoping for hostel=cheap, you might be in for a disappointment, Barcelona is such a popular destination that prices are never "low", anytime during the year, much less in July.

Free Hostal on Londres. i’ve stayed there multiple times and loved it.

Years ago, I stayed at Hotel de Cuatro Naciones on Las Ramblas, an historically interesting budget hotel that still has lower prices for Barcelona. The building goes back to the 1700s. Photos of famous guests when it was more glamorous include Buffalo Bill and his wild west show, Ulysses Grant and Albert Einstein..

We've stayed in Casa Gracia Barcelona a few times, and really liked it. It's in Eixample. I don't know about availability currently, but one we booked in February less than a week ahead and got a room.