Banks, who passed away aged 81 earlier this month, played 194 times for Stoke between 1967-1973 – winning the club's only major trophy to date, the League Cup in 1972.
Sam Vokes gave Stoke the perfect start, opening the scoring with an improvised close-range finish with just five minutes played.
Adomah earned Villa a share of the spoils after the hour when he hit a well-struck drive from inside the penalty area after a swift counter-attack.
Nathan Jones' hosts were ahead after five minutes when Sam Clucas' low cross from the left was met by Vokes, who backheeled left-footed in the six-yard box to see the ball cross the line despite goalkeeper Jed Steer's best efforts.
Both teams were then forced in to early changes after injuries, with goalscorer Vokes making way for Mame Biram Diouf, while Adomah replaced full-back Alan Hutton for Villa.
In pursuit of a quick-fire second goal, midfielder Charlie Adam, making his return to the Stoke side, tried his luck from range, seeing his drive from 30-yards dip wide of Steer's post.
With the clock ticking down until half-time, captain Danny Batth later saw his powerful header following Adam's corner from the left well held by Steer.
Pushing for a route back in to the contest in Staffordshire, Villa had their only real chance of the opening half when Tammy Abraham fed Anwar El Ghazi, but the Dutch midfielder could not get his right-footed attempt on target and could only shoot over the crossbar.
After the break, Villa began the second-half brightly, despite forcing several set-piece situations and seeing referee David Webb wave away a penalty appeal when Tammy Abraham was brought down inside the area.
Up at the other end, Stoke looked to extend their lead in front of an expectant home support, with Tom Ince firing a snapshot straight in to the grateful grasp of Steer.
But Dean Smith's visitors were level after 62 minutes after El Ghazi broke at pace down the left to feed the on-rushing Adomah, who thumped his low strike beyond the dive of goalkeeper Jack Butland to find the bottom-right corner in front of a packed away end.
And within two minutes of drawing level, the buoyant Villa almost went in front after Kortney Hause crossed for Abraham, who connected with the ball to force Butland to claw his goal-bound chance away to safety.
In the closing stages, Conor Hourihane sent his left-footed free-kick from 20-yards narrowly over Butland's upright, as both sides had to settle for a point.