Yeah, District 1 is definitely in the centre - we stayed at a place called Aries Ben Thanh Central Hotel which was about 180 for 6 nights (you could totally go cheaper though). It was down a back street but near the best Banh Mi cart in the city (Banh Mi 37 Nguyen Trai).
You could stay out of District one to save cash as Uber is an incredibly accessible way to get around (a couple of quid for a journey, or go for a... Lyft, I think it was called, on the backs of mopeds). We chose these as rainstorms can screw your day up very easily.
If you like a beer then Pasteur Street Brewing Company is very popular with ex-pats and is worth a visit.
Definitely worth going for a massage somewhere too, as again they're very cheap (read: reputable massage places - see google reviews to avoid the seedy places).