Addison, 26, was a free agent after leaving Bournemouth in the summer, having spent much of last season on loan at Scunthorpe and Blackpool.
He only made 22 appearances in three years with the Cherries after joining from boyhood club Derby in 2012.
Samuelsen, 18, will be on loan with Posh until 2 January.
The Norway-born teenager made two substitute appearances for the Hammers in their Europa League tie against Birkirkara in July.
Addison, meanwhile, will provide cover for the injured Gabriel Zakuani and Shaun Brisley, and Jack Baldwin and Ricardo Santos, who are suspended.
He was named in the starting line-up for Tuesday's home game against Barnsley, with Samuelsen on the bench.
League One side Peterborough United have signed centre-back Miles Addison on a short-term contract and West Ham midfielder Martin Samuelsen on loan.