The selection of Neilson allows Brady to avoid a nasty primary fight between two Democrats.
Neilson is finishing his first term in a district that was moved by redistricting in 2012 from Northeast Philly to York County.
He was planning to run against state Rep. John Sabatina Jr., who is seeking a fifth term.
Neilson worked for Local 98 of the International Brotherhood of Electrical Workers. Sabatina is the son of a longtime Northeast Philly ward leader.
Local 98 chief John Dougherty, who voted as the 1st Ward leader, dismissed the notion that he is trying to build his union's power in Council. Neilson, like Councilman Bobby Henon, previously served as the union's political director.
"It has nothing to do with their history with me," Dougherty said. "They're just really good people who came up the line."
Neilson, too, disputed that he would be part of a union block in Council, although he called himself a "Local 98 guy" and said he spoke often with Dougherty.
"He made it my decision," Neilson said of the Council run.
Former Councilman Bill Green IV resigned from Council last month to become chairman of the School Reform Commission, creating the vacancy.
Matt Wolfe, a Republican ward leader from West Philly, is seeking his party's nomination next week for the special election.
Independent candidates must collect 1,785 petition signatures from registered voters by April 8 to win a spot on the ballot.
The Democratic primary challenges filed yesterday include:
* State Rep. J.P. Miranda, arrested in January for an alleged ghost-employee scheme used to funnel state money to his sister, is challenged in North Philly's 197th District by Leslie Acosta, Danilo Burgos, Ben Ramos, Edward Lloyd Jr. and Juan Rodriguez.
* Former state Rep. Babette Josephs is challenging state Rep. Brian Sims, who defeated her in the 2012 primary in Center City's 182nd District.
* State Sen. Tina Tartaglione is challenged in the 2nd District, stretching from Kensington to Fox Chase, by Dan Savage and Tomas Sanchez.
* State Sen. LeAnna Washington is challenged in the 4th District, stretching from Logan to Willow Grove in Montgomery County, by Brian Gralnick and Arthur Haywood III.
* State Sen. Anthony Hardy Williams is challenged in West Philly's 8th District by Christopher Broach.
* State Rep. Kevin Boyle is challenged in Northeast Philly's 172nd District by Jeffrey Voice.
* State Rep. Mike O'Brien is challenged in the 175th District, which stretches from Pennsport to Kensington, by Sean Sullivan.
* State Rep. James Clay Jr. is challenged in the 179th District, stretching from Olney to Frankford, by Jason Dawkins and David Lee Hall.
* State Rep. Angel Cruz is challenged in the 180th District - which includes North Philly, Kensington, Feltonville and Juniata Park - by Quetcy Lozada.
* State Rep. W. Curtis Thomas is challenged in North Philly's 181st District by Emily Rodriguez, Kenneth Walker Jr. and Kenneth Williams-Medley.
* State Rep. James Roebuck is challenged in West Philly's 188th District by Algernong Allen.
* State Rep. Vanessa Lowery Brown is challenged in the 190th District, stretching from West Philly to Wynnefield Heights, by Wanda Logan and Isaac Patterson.
* State Rep. Pam DeLissio is challenged in the 194th District, stretching from East Falls to Roxborough, by David Henderson and Sean Stevens.
* State Rep. Rosita Youngblood is challenged in the 198th District, which includes Nicetown, Logan and Ogontz, by John Lee Connelly.
* State Rep. Cherelle Parker is challenged in Northwest Philly's 200th District by Bobbie Curry.
* State Rep. Stephen Kinsey is challenged in the 201st District, which includes Logan, Fern Rock and Germantown, by Lamont Thomas.
* State Rep. Mark Cohen is challenged in Northeast Philly's 202nd District by Jared Solomon.
On Twitter: @ChrisBrennanDN