Miguel Herrera has been fired as Mexico coach following allegations that he punched a television reporter.
Former Arsenal midfielder Abou Diaby has rejected an offer to join West Brom in favour of finding a new club in France.
The deal will see the 35-year-old long-throw specialist feature in a seventh season at the Britannia Stadium.
Delap signed for the Potters in a £500,000 move from Sunderland, was part of the 2007/08 promotion-winning side and has gone on to play over 200 matches for the club.
Manager Tony Pulis told stokecityfc.com: "Rory deserves this deal. It is important we look after those players who have been here since the start of this journey.
"It's not just what he does on the field; he's been an inspiration off the pitch as well.
"He has played close to 600 games during his playing career, and the vast majority of those have been in the top-flight and so he has offers so much more than just a long throw.
"He is the ultimate professional and it is important we look after certain players at the football club who started all this off."
Delap was also delighted to agree new terms.
"I'm chuffed to be honest," he said. "I wasn't sure what was going to happen this summer, so to get it all sorted out now, with the rest of the season to come, is fantastic for me.
"I've made no secret about the fact that I love it here at Stoke, and to have been part of the success we have enjoyed over the past couple of years has been unbelievable.
"I still feel there is more to come though, and hopefully I can play my part in ensuring that proves to be the case."The first chance fell to Stoke but Robert Huth's header from a Delap long throw bounced harmlessly into the hands of Tim Howard.
Sylvain Distin headed Leon Osman's corner wide at the other end and Anichebe charged down Asmir Begovic's clearance but was well wide of goal and the ball bounced away for a goal kick.
Distin made an important clearance as Jon Walters looked to meet Matthew Etherington's deep cross. The pair clashed heads in the process, with Walters requiring touchline treatment and re-emerging heavily bandaged.
Etherington dinked an effort over after Jones' knockdown and when another Delap launch reached the far post, Huth headed over.
Anichebe landed awkwardly after leaping for a header and, after attempting to continue, was replaced by Magaye Gueye. Anichebe, who has had a long history of injury problems, departed tearfully for the tunnel.
Howard did enough to prevent Jones reaching Delap's next throw before another was unconvincingly scrambled out of the six yard box.
HT: Stoke 0 Everton 1 Everton threatened from Steven Pienaar's cross but Marouane Fellaini was unable to dig out a shot and Osman's effort was easy for Begovic.
Delap was at his menacing best from the sideline and United States keeper Howard resorted to a basketball slam-dunk over the bar as Ryan Shawcross closed in on Huth's header goalwards.
Nikica Jelavic instinctively met Gueye's awkward bouncing cross with his knee but the effort cleared the crossbar.
Pienaar flashed an effort just over, Begovic credited with a touch, and when the ensuing corner was not cleared Gueye drove powerfully wide of the far post.
If Jelavic's earlier effort was unorthodox, it had nothing on Everton's opener moments before half-time.
Tony Hibbert, whose crosses caused such problems for Manchester United during the remarkable 4-4 draw at Old Trafford, delivered once more and after Tim Cahill failed to connect properly, defender Marc Wilson headed into the ground and the ball bounced in off the side of Peter Crouch's head for an unfortunate own goal.
INTRO: Peter Crouch's freak own goal gave Everton a half-time lead over Stoke at the Britannia Stadium
Tim Cahill made minimal contact with Tony Hibbert's cross and Marc Wilson's subsequent header bounced up into the face of the sometime England striker and past keeper Asmir Begovic.
Stoke had threatened via Rory Delap's long throws but Everton came into the game as the half progressed and Nikica Jelavic and Steven Pienaar both went close before the bizarre 44th-minute breakthrough.
The second half started quietly until Everton appealed for handball when Osman's ball into the box struck Huth. The appeals were rightly waved away, though referee Anthony Taylor returned to book Wilson for an earlier foul on Pienaar for which he had played advantage.
Stoke also failed to win a penalty after a more convincing handball appeal against Hibbert, and again when Jones fell under Osman's aerial challenge.
In between those two incidents Cahill failed to connect with a volley when Distin's long throw - not up to the Delap standard - was half-cleared.
Stoke boss Tony Pulis deployed his three substitutes in one fell swoop as Cameron Jerome, Ricardo Fuller and Glenn Whelan replaced Walters, Crouch and, surprisingly, Delap.
And the former made an immediate impact as he controlled the ball on halfway, bundled his way through the defence and finished from 18 yards with the aid of a deflection off Phil Jagielka.
FT: Stoke 1 Everton 1 Fuller did superbly to retain possession under pressure from three Evertonians and produced more trickery on the end of the same move to win his side a corner, which was cleared unconvincingly.
Huth and Jelavic were spoken to after an off-the-ball clash before Everton substitute Gueye was himself replaced by Denis Stracqualursi.
Osman's deflected strike from the edge of the box bounced tamely into Begovic's arms before Fuller, tracking back to cover the run of Hibbert, cleared Jelavic's cross.
Jelavic could not connect cleanly with Distin's cross but the best chance of a winner fell to Stoke in stoppage time, when Whelan's corner led to a scramble which ended with Jones blazing over.
Howard then did well to save Huth's deflected drive, from a free-kick after Pienaar was booked for a cynical foul on Fuller.
Intro: Substitute Cameron Jerome made an immediate impact with the goal which earned Stoke a draw at home to Everton in tonight's Barclays Premier League clash.
Peter Crouch's freak own goal a minute before half-time put Everton ahead, but Jerome equalised with his first contribution 90 seconds after being introduced as part of a treble change.
It was a deserved point for the Potters, who carried a first-half threat from Rory Delap's long throws and improved their all-round performance after the break, and they could even have won it in stoppage time through Kenwyne Jones or Robert Huth.