The 33-year-old made 19 appearances for the Seagulls last term after joining on loan in January, helping them finish third, before a play-off semi-final defeat by Sheffield Wednesday.
Sidwell played only four games for Premier League club Stoke last season.
"He knows exactly what the Championship is all about and what it takes to get promoted," manager Chris Hughton said.
"He was excellent for us in the run-in, and was a key figure on and off the pitch as one of the senior members of the squad."
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Justice Secretary Michael Gove rejected a ruling last month by the Parole Board that the notorious gangster was suitable for a move to open conditions.
Noye, 68, was jailed for life for the road rage murder of 21-year-old Stephen Cameron in 1996 during a fight on an M25 slip road at Swanley in Kent.
He went on the run after the killing, but was captured in Spain in 1998.
He was jailed in 2000 with a minimum term set at 16 years in 2002 by the then Home Secretary David Blunkett.
The parole board considered a request from Noye to be released from Wayland Prison in Norfolk.
Being moved to an open prison is the first stage of the process that usually leads to release.
Such an intervention by the Secretary of State is rare, with fewer than 1% of Parole Board recommendations being turned down over the last five years.
About 40 recommendations out of up to 6,000 from 2010 to this year have been rejected by the Justice Secretary.
Noye is entitled to another parole hearing within two years.
After the Parole Board made its recommendation Mr Cameron's father Ken said he and his wife Toni believed he would abscond from an open prison.
"Finally we have got justice again for Stephen,"  he said.
"He's going to be there for another two-and-a-half years with a bit of luck."
Noye became one of Britain's most notorious criminals through his involvement in the £26m Brink's-Mat raid in 1983 - one of the UK's biggest robberies.
Six armed men posed as security guards and stole 6,800 gold bars from a warehouse at Heathrow Airport.
During the police investigation, undercover officer Det Con John Fordham was stabbed to death in the grounds of Noye's mansion in West Kingsdown, Kent in 1985.
Noye was cleared of murder on the grounds of self defence, but jailed for 14 years for handling stolen bullion.

Members of the National Union of Teachers (NUT) said it was to highlight wider problems including teacher redundancies.
Powys council's cabinet member for schools, Arwel Jones, said he was "surprised and disappointed".
The NUT said it represented about half of teachers in Powys and 83.5% of those balloted voted in favour of action.
The council is currently consulting on plans to reorganise secondary schools.
Mr Jones added: "Strike action will not deflect our programme but could disrupt the work of school pupils at a crucial time."
Championship side Brighton have signed midfielder Steve Sidwell on a one-year deal following his release by Stoke.
[NEXT_CONCEPT]
Road rage killer Kenneth Noye has been refused a move to an open prison over fears for public safety.
[NEXT_CONCEPT]
Teachers in Powys are set to go on strike on Thursday in protest over "excessive workload and cuts".