The phrase Benjamin’s portion, also Benjamin’s mess¹, denotes the largest share—cf. origin of ‘the lion’s share’: ‘le partage du lion’.

¹ Here, mess denotes a portion, a serving—cf. meaning and origin of the phrase ‘mess of pottage’.

This phrase alludes to the Book of Genesis, 43:34; Benjamin, the patriarch Jacob’s youngest son, receives the largest portion of food when he and his brothers are entertained by their long-lost brother Joseph, whom they have encountered in Egypt but have not recognised:

from the New International Version (2011):
When portions were served to them from Joseph’s table, Benjamin’s portion was five times as much as anyone else’s. So they feasted and drank freely with him.
[Holy Bible, New International Version®, NIV® Copyright ©1973, 1978, 1984, 2011 by Biblica, Inc.® Used by permission. All rights reserved worldwide]

The earliest instance of the phrase in the Oxford English Dictionary (2^nd edition, 1989) dates from 1840, but I have found much earlier uses. The phrase seems to have appeared as Benjamin’s portion, and the earliest occurrence that I have found is from the following paragraph published in Harrop’s Manchester Mercury, and General Advertiser (Manchester, Lancashire, England) of Tuesday 6^th February 1753—this newspaper does not clearly state to what and whom it refers:

The profitable Place said, in our News Papers, to be given to a certain great and worthy Person, who lately resigned, is held only in Trust; and is said to be, amongst the Connoisseurs in Custom-House Literature, Benjamin’s Portion; the Reward of his tedious and tiresome Journies, and of his Negotiations without End.

The second-earliest instance of Benjamin’s portion that I have found is from Extract from the Preface² to Sixteen Sermons […] by the Worthy and eminently Pious Christopher Love³, published (as part of proposals for printing by subscription the sermons in question) in The Pennsylvania Journal, and the Weekly Advertiser (Philadelphia, Pennsylvania) of Wednesday 8^th February 1792:

If thou art one of those, that upon good Scripture-grounds canst say thou hast the truth of Grace, labour then after Growth in Grace. No Christian should content himself with any measure of Grace attained, for he is like to make use of all the Grace he hath, had he a Benjamin’s portion.

² This preface was signed by Edmond Calamy, Simeon Ashe, Jer. Whitaker and William Taylor.
³ Christopher Love (1618-51), a Welsh Presbyterian preacher, was executed for plotting to restore Charles II to the throne.

The earliest occurrence of Benjamin’s mess that I have found is from the speech introducing a motion for inquiry into the distress of the industrious classes that Edward Davies Davenport (1778-1847), MP for Shaftesbury, Dorset, delivered at the House of Commons on Thursday 14^th June 1827, and that several London newspapers published the next day—the following, for example, is from The Courier of Friday 15^th June 1827:

[Mr. Davenport] added, it was his intention, if no other Member were found to do it, to call the attention of the House to the subject of the salaries to public officers, because, if it were determined to keep the country on short allowance, it would be but justice to let all persons share alike, and to let no one enjoy a Benjamin’s mess.

In the following paragraph from The Bradford Daily Telegraph (Bradford, Yorkshire, England) of Monday 19^th December 1887, the phrase, used as the title, punningly denotes the opposite of what it usually does:

Benjamin’s Portion.—Benjamin Rushton, a notorious pauper, residing at the Workhouse, was placed in the dock at the Borough Police Court, Halifax, on Saturday, for the thirty-sixth time. On Friday the dinner at the Workhouse consisted of bread and cheese. Benjamin’s portion of the cheese did not suit him, so he threw it at the head of Thomas Laycock, the server of the victuals, and then indulged in profane language.—Robert Billington, the new master, told his Worship (Ald. Riley) that on the day in question he had occasion to remonstrate with the prisoner for using bad language. As usual Benjamin became violent, and it took four of the officers to carry him out the master’s presence. Prisoner complained of being always served with the “worst cuts.”—Ald. Riley, who said he knew the prisoner of old, committed him to gaol for seven days.