The 24-year-old, who is the brother of former Manchester United defender Wes Brown, had been on trial with the League One club and featured for their Under-23s earlier this week.
Brown, who left Bury on a free transfer in the summer, has earned himself a deal until the start of 2017.
"He comes in and gives us cover at right back or right wing-back. He adds weight to our squad and knows the division well, Sheffield United boss Chris Wilder told the club's official website.
"He has done everything we have asked of him, trained well and impressed. His pedigree is good and this is an opportunity to earn a longer deal."
The defender started his career with Manchester United and has also represented Watford and Barnsley, in addition to five separate loan spells during his time at Old Trafford.