The machine, which was used in monthly prize draws from the scheme's launch in 1957 until 1972, will take its place in the History of Computing Gallery on June 26.
The height of technology at the time, the public flocked to see Ernie (Electronic Random Number Indicator Equipment) in action at the Premium Bonds headquarters in St Annes.
Jack Armitage, 77, who was the engineer in charge of Ernie for 16 years, said: "We had it painted gold and in the background there was this music which was like something from Doctor Who. It seems really corny now, but everyone loved it and wanted to have their photo taken with him."
Alongside Ernie will be photographs from the launch of Premium Bonds in 1956, original posters advertising Bonds and items reflecting the immense popularity of the computer, including cards and poems sent to the random number generator by the British public.
Tilly Blyth, curator of computing at the Science Museum, said: "This is a very important machine in the history of computing in Britain, as well as being one of the most-loved.
"Now, we see computers as an invaluable part of our daily lives, but it was Ernie that helped us become confident in the use of digital electronic machines."
Ernie 1, which weighs 1.8 tonnes and is the size of a van, has been since been replaced by a succession of smaller and faster models. Ernie 4, which is currently in use, is no bigger than a PC.