This was determined by the Advanced Service Rating Score, a combination of length of service, length of time overseas, awards earned, and number of dependents. I talked about it earlier here, but I'll paste it below.

The "points system" you are describing is very historically accurate. Technically, it was called the "Advanced Service Rating score" and decided which troops in the European Theater of Operations would be sent home as part of Operation Magic Carpet, the mass demobilization as World War II drew to a close, or be retrained for the invasion of Japan.

In immediate postwar Europe, U.S. units were divided into four categories, with men transferred in and out of them as needed. Empty slots left by men sent home because of points were to be filled by new replacements.

Category	Explanation
I	Units to remain in Europe as occupation forces
II	Units to be redeployed to the Pacific for the invasion of Japan. Some soldiers were to go directly there (the 86th Infantry Division, having fought in Europe for only 34 days, was guarding POW pens in the Philippines on V-J Day!), while others were to return to the United States for a training period before deploying
III	Units to be reorganized and retrained before being classified as Category I or II
IV	Units to be sent home and inactivated with personnel discharged; essentially "vehicles" full of men eligible for discharge

In order to be sent home, a man needed to accumulate 85 points as you have noted. Many soldiers earned well over 140 points; this just rounded down to 85. No points were given for age or marital status.

Criteria	Points
1 month of military service	1 point
1 month of military service overseas (in addition to total time in service)	1 point
Combat award (medal or campaign streamer)	5 points
Having a dependent under the age of eighteen	12 points per dependent

Men who had three or more dependents under eighteen were eligible for immediate repatriation and discharge regardless of the amount of points they had earned.

As an example, a man who served for 3 years in the Army, including 1 and a half years overseas, had a child under 18, and fought in the Northern France, Ardennes-Alsace, Rhineland and Central Europe Campaigns, earning a Silver Star and Purple Heart, would have:

36 (months in service) + 18 (months overseas) + 12 (dependent under 18) + 5 (Northern France) + 5 (Rhineland) + 5 (Ardennes-Alsace) + 5 (Central Europe) + 5 (Silver Star) + 5 (Purple Heart) =

91 points, enough to go home!

Category I, II, and III units would contain no men who had earned 85 or more points, while Category IV units would exclusively be made up of men who had earned 85 or more points. The units to be shipped to the Pacific were to contain those troops "least eligible" for discharge. Many veteran divisions that had not yet returned home were whittled to absurdly low levels by the reduction of the ASR score down to 75 points (and resultant transfers of personnel) in September 1945; the 88th Infantry Division was down to only 1,200 men! There were also gripes that the system unfairly rewarded rear-echelon troops; infantrymen didn't get additional points for risking their lives daily, besides points for medals, which all troops could get.

Campaign Streamers in chronological order:

Campaign	Dates
Egypt-Libya	June 11, 1942 - February 13, 1943
Air Offensive, Europe	4 July 1942 - 5 June 1944
Algeria-French Morocco	8 - 11 November 1942
Tunisia	17 November 1942 - 13 May 1943
Sicily	9 July - 17 August 1943
Naples-Foggia	9 September 1943 - 21 January 1944
Anzio	22 January - 24 May 1944
Rome-Arno	22 January - 9 September 1944
Normandy	6 June - 24 July 1944
Northern France	25 July - 14 September 1944
Southern France	15 August - 14 September 1944
North Apennines	10 September 1944 - 4 April 1945
Rhineland	15 September 1944 - 21 March 1945
Ardennes-Alsace	16 December 1944 - 25 January 1945
Central Europe	22 March - 11 May 1945
Po Valley	5 April - 8 May 1945