This is just what I think the ship's history could be, as the canon of
Star Trek¨ doesn't have much on it. 

Earth was facing a challenge, to build a starship capable of
accelerating to Warp Five, which was almost unattainable.

That is until Matthew Robert Harrison of Macon Georgia, told Zephram
Cochrane, that he'd taken pictures of the interior of a Vulcan ship.
Zephram and his crack team of scientists quickly made observations and
notes about the technology Harrison had taken pictures of. Zephram hired

Harrison to get some more information about the Vulcan ship. Harrison
used a handheld sensor and penetrating imaging device to scan the
engineering section of the ship. After paying a large sum, Cochrane and
his team used the information to draw up some blueprints.

After a construction time of one year, three weeks and two days,
Enterprise was complete. During a testing run, the unexpected feature of
the Vulcan design was discovered. Enterprise was towed out of spacedock
and powered to one-quarter. Instantly the ship went to Warp 3.2!  The
ship used substantially less power than a ship with Cochrane nacelles.
The ring made the ship go faster at a forth of the energy requirements!
Enterprise was placed on duty as a galactic explorer, even though it
never got past Warp 3.9.

Two months out of spacedock, the crew reported seeing strange things
onboard the ship. Many of the crew fell ill onboard the vessel.
Enterprise was ordered back for immediate testing. Upon returning to
Earth, they found Enterprise was emitting dangerous radiation from it's
warp rings, apparently Harrison had missed a vital part of the design.
Most of the crew died of radiation poisoning while on Earth, among them
was Henry Archer's brother Robert. After repairs and an overhaul to fix
most of the radiation problem, Enterprise was placed back on duty as a
solar patrol ship as she could no longer use her warp rings for fear of
another radiation disaster. She finally met her end in a blaze of glory,
by scuttling in the atmosphere after serving Earth for fifty years.

Henry made it his lifelong passion to make a Warp Five ship, and he
did, they also named it Enterprise. Henry didn't make it long enough to
see his ship though. His son, Captain John Archer now commands Henry's
ship: NX-01.