The 24-year-old, who played 16 games on loan at Kilmarnock last season, will compete with James Tavernier.
Hodson started his career at Watford and played 51 games in three seasons at MK Dons after a loan at Brentford.
He joins new faces Matt Gilks, Clint Hill, Joey Barton, Niko Kranjcar, Jordan Rossiter, Matt Crooks and Josh Windass at Ibrox.
Rangers paid an undisclosed fee for Hodson, who was part of Northern Ireland's squad at Euro 2016, but did not feature in any of their four matches.

England's win over South Africa on Sunday secured the Twenty20 series.
The game was officially sold out, as were England's two ICC Champions Trophy matches at the SSE Swalec Stadium.
The near-15,000 crowd comes as counties bid for the next round of five years' worth of international games.
"To have three full houses over 19 days for England matches is something we've very proud of," said Morris.
Cardiff hosted England's win over New Zealand and semi-final defeat to Pakistan in the Champions Trophy, along with Sri Lanka versus Pakistan and the less well-attended Bangladesh v New Zealand match.
Around 3,000 seats were not occupied for the England v New Zealand match because of poor weather forecasts, while cancelled trains are believed to have contributed to some empty seats for the T20 on Sunday.
But a rapid re-allocation of tickets worked well for the England-Pakistan game after fears that India supporters, who bought tickets for both semi-finals, would not turn up.
"It's particularly important (to have big crowds) this year, because at the back end of the year we are likely along with other counties to be bidding for our international programme between the years 2020 and 2024," explained Morris.
"There's also the new city-based franchise T20 as well, so it's important we can demonstrate we can fill houses and people have a great time when they are here."
England's score of 181 in 20 overs against South Africa could also come as a relief, after captain Eoin Morgan criticised the use of a previously-used wicket for the Champions Trophy semi-final, a choice attributed to the International Cricket Council.
"I wasn't surprised (with the score), it's a wicket that has played extremely well this year in domestic and international cricket, in recent years we've had some very good marks," said former county captain and England opener Morris.
Glamorgan have forecast an operating profit for 2017 on the back of five days of international cricket.
But it appears unlikely they will add another overseas player to their squad for the T20 Blast, after signing South African paceman Marchant de Lange on a three-year deal for all forms of cricket.

Midfielder Jeff Hendrick has been ruled out because of a shoulder problem and has returned to his club Derby County.
Harry Arter will miss the Dublin games because of an Achilles injury.
The Bournemouth player, who has only one Republic cap, has returned to his club for treatment.
Stoke City forward Walters sat out training sessions in Dublin over recent days.
"Jeff has got a shoulder injury that he has had to deal with for quite a number of weeks now so we thought it best going back and getting is sorted a club level," said Republic manager Martin O'Neill.
"Jon, I think he will be fit for Tuesday, he just thought he might be best to go home for a day or two. He should be back on Friday night to watch the game. He will definitely play on Tuesday.
"Robbie Keane sat out training again on Wednesday but there is no real problem. He will be okay and will be available for any of the two games."
Republic of Ireland Euro 2016 fixtures (all times BST)
Monday, 13 June
Republic of Ireland v Sweden (17:00; Stade de France, Paris)
Saturday, 18 June
Belgium v Republic of Ireland (14:00; Stade de Bordeaux)
Wednesday, 22 June
Italy v Republic of Ireland (20:00; Stade Pierre-Mauroy, Lille)
Rangers have made right-back Lee Hodson their eighth summer signing on a three-year deal from MK Dons.
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Glamorgan chief executive Hugh Morris says Cardiff's third sell-out for an England match in June will boost their hopes of attracting further international matches.
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Republic of Ireland striker Jonathan Walters will not feature in Friday's friendly with Switzerland because of a hamstring injury, but should be fit to face Slovakia on Tuesday.