The 20-year-old, the son of former Hammers, Newcastle and England midfielder Rob Lee, had a loan spell at Luton's League Two rivals Southend United earlier this season.
But he was injured just 95 minutes into a training session with the Shrimpers and returned to Upton Park.
Lee has made two substitute appearances for West Ham and was on loan at Colchester United last season.
Southend boss Phil Brown had wanted Lee to return to Roots Hall, but he has instead joined their fellow promotion contenders.
Luton Town have signed West Ham striker Elliot Lee on loan until 24 March.