Anywhere inside the Circle Line is fine (as, indeed, are plenty of locations just a little further out). Much of the cheaper accommodation in London is located around Kings Cross, Paddington/Bayswater and Stratford but there also some reasonably-priced hotels around Bloomsbury, Kensington and elsewhere.
To give you an idea of how those locations fit in with your plans:
Kings Cross is within walking distance of Euston. It's about 15 minutes from Hyde Park on the Tube.
Paddington & Bayswater (which are neighbouring areas) are about 10 minutes on the Tube from both Euston and Hyde Park.
Stratford is the only area I've suggested which is outside the Circle Line. It would take about 30 minutes on the Tube to get to Hyde Park and about the same to get to Euston.
Bloomsbury to Hyde Park might take around 15 minutes or about 10 minutes to Euston.
You can walk from Kensington to Hyde Park and get to Euston on the Tube in under 20 minutes.
For decent quality budget accommodation try this website:
http://www.hostelworld.com/
Don't be put of by the word 'hostel' in it's name. Conduct an intitial search but then use the filters to see just hotels and B&Bs. Read the reviews carefully and look for places in the areas I've suggested.