The 23-year-old will replace flanker Allister Hogg who starts at number eight in place of Simon Taylor, who is recovering from surgery on his knee.
Brown has three Scotland caps after making a try-scoring debut in Romania last summer following a non-cap appearance in the win over the Barbarians in Aberdeen 12 months ago.
But Hadden will maintain a strong starting line-up this week as he prepares for the two-Test tour of South Africa next month.
Scott Lawson of Glasgow Warriors will start at hooker alongside props Gavin Kerr and Craig Smith of Edinburgh Gunners, who is included ahead of Bruce Douglas.
Hadden insists his team selection is indicative of the importance he has placed on the game against the Barbarians.
The Scotland boss has vowed to use the summer matches as the start of his preparations for the World Cup next year.
Hadden said: "We regard the contest against the Barbarians very much as a Test match.
"We are looking to develop some of our attacking repertoire but most of all we're looking to put in a performance.
"The Barbarians game is important both relative to the tour of South Africa and to next year's World Cup as we only have around a dozen games to prepare for the World Cup."
And Hadden is confident both Smith and Brown will prove their worth to his plans on Wednesday.
He added: "Craig has done well when we've introduced him from the bench for his 20-minute flourishes. He deserves the opportunity to start.
"Kelly was in our travelling squad in both Wales and Italy during the Six Nations Championship.
"He has played extremely well for the Borders this season and I need to see him perform at this level."
Scotland team to play the Barbarians at Murrayfield on Wednesday, May 31 is: H Southwell (Edinburgh Gunners); C Paterson (Edinburgh Gunners), M Di Rollo (Edinburgh Gunners), A Henderson (Glasgow Warriors), S Lamont (Northampton Saints); D Parks (Glasgow Warriors), M Blair (Edinburgh Gunners); G Kerr (Leeds Tykes), S Lawson (Glasgow Warriors), C Smith (Edinburgh Gunners), A Kellock (Edinburgh Gunners), S Murray (Edinburgh Gunners), J White (Sale Sharks, capt), A Hogg (Edinburgh Gunners), K Brown (Border Reivers).
Replacements: D Hall (Edinburgh Gunners), B Douglas (Border Reivers), S MacLeod (Border Reivers), D Macfadyen (Glasgow Warriors), S Pinder (Glasgow Warriors), G Ross (Leeds Tykes), S Webster (Edinburgh Gunners).