Brown, a North Carolina alumnus, told ACC Network's Packer and Durham on Friday that he believes Williams deserves the chance to choose his successor and that the next coach should already have Tar Heels ties.
"In my gut, I hope Roy Williams picks his successor, and I hope it's someone from the Carolina family," Brown told Packer and Durham. "... I know what Coach [Dean] Smith would want, I know what Coach [Frank] McGuire would want. ... Keep it in the family, and let's move forward."
Williams announced his retirement Thursday after 18 seasons at North Carolina and 33 seasons as a college coach. After a Hall of Fame career that included more than 900 wins and three national championships with the Tar Heels, Williams said that "I no longer feel that I am the right man for the job."
Athletic director Bubba Cunningham is tasked with figuring out how the Tar Heels' search will go, as the program seeks only its sixth coach since 1952.
"I believe we've got the best job in college basketball," Cunningham said. "This job doesn't come open very often. We need to spend a great deal of time thinking about who is the right person right now."
Cunningham said he and chancellor Kevin Guskiewicz would handle the process instead of using a search firm or committee. He said he has sought input from numerous former players and from Williams, who Cunningham said had already shared his opinion. Williams also mentioned giving his input during his farewell news conference.
The athletic director said that he prefers someone with head-coaching experience but that it isn't required.
"The history and tradition here is winning," Cunningham said. "We've had it in the family for a long period of time. That is important. But it's not the only factor in trying to make a decision like this."
Hiring someone with North Carolina ties is also a known factor. That worked with longtime assistant Bill Guthridge taking over after Smith's 1997 retirement and with Williams' return in 2003, but it didn't work during the tumultuous tenure of Matt Doherty after Williams turned down the job following Guthridge's retirement in 2000; Brown, who was the Philadelphia 76ers' coach at the time, had also expressed an interest in coaching North Carolina.
"I just want Roy to have a say in this thing," Brown told Packer and Durham. "And I know, no matter who takes over, if Roy has touched his life, if Coach Smith has touched his life, we'll be in good hands."
There are experienced coaches with strong program connections, such as UNC Greensboro's Wes Miller, who played on Williams' 2005 NCAA championship team; Stanford's Jerod Haase, who coached under Williams at North Carolina; and Hubert Davis, who played for Smith and has spent the past nine seasons on Williams' staff.
Cunningham said there is no specific timetable to make a hire.
Brown has recently been linked to a role at Indiana, with sources telling ESPN's Adrian Wojnarowski last month that new Hoosiers coach Mike Woodson has talked with Brown about a possible on-campus, non-recruiting role.
The Associated Press contributed to this report.