With a decent bar selection that serves until late? Say after midnight? Preferably not too far from a Wetherspoons also.
We often stay at Travelodge The Strand as it is well located being across the road from Albert Dock and a short walk to all the bars of the city centre. There is a Spoons (The Captain Alexander) around 2 mins walk on James Street.
The hotel is very clean, there is a bar, and a cafe area where they serve meals, snacks and breakfast. The rooms overlooking the Strand can be noisy but with the window closed you can't hear much road noise.
There are plenty, Premier Inn, Travelodge,Ibis, Lord Nelson and loads more.
It depends when you’re going as when Liverpool FC are playing hotel prices treble or even more, especially on a Saturday night.
I think it depends on when you want to visit as to what is 'budget'.
Groupon have deals, but when football or Taylor Swift is in town then prices are expensive