Is the phrase, “Let ‘em up easy” Abraham Lincoln’s one-off phrase or an obsolete idiom?

It's a bit of a pun. While the phrase let them up easy may be rare, the phrase let him down easy is not.

Let him down easy could refer to a man who is fainting – catch him, and lay him gently on the ground – but it could also refer to an emotional blow. For example, one might advise a woman who intends on breaking up with her boyfriend, a soldier who has been deployed for six months, “Don't just break up with him the moment he gets of the plane – let him down easy.”

At the end of the Civil War, many in the Union were still bitter toward their enemies. Lincoln, however, was a proponent of reconstruction; he felt that the Southern states needed to be gradually welcomed back into the Union in a non-punitive manner.

I've not heard the quote before, but it sounds Lincolnesque. Lincoln was rather fond of quips and puns; this one is attributed to him:

The man who can't make mistakes, can't make anything.

The phrase “Let ’em up easy” probably is drawn from a wrestling or rough-and-tumble fighting context. With the opponent pinned down, or with the loser underfoot, the winner has a choice: extract onerous concessions or promises from the loser, or let them up with some dignity still attached.

Fans of Lincoln are familiar with the line and likewise know that Lincoln was an avid wrestler as a youth and young man. Given that background, it is a great line and likely easily understood by the general hearing the directive.

It's sort of a first cousin to "don't squeeze the last nickel out of the deal," meaning when you really have someone over a barrel while negotiating, don't make quite has harsh a deal as perhaps you could.

Looking into the history of this anecdote, I found that it is recorded (in somewhat different form) in Asa Bartlett, History of the Twelfth Regiment, New Hampshire Volunteers in the War of the Rebellion (1897). Bartlett's source is a letter to the Boston Globe newspaper, dated April 26, 1885, by Thomas Thatcher Graves, who 30 years earlier was aide-de-camp to Union General Godfrey Weitzel (the general in the anecdote) and who claims to have been present when the exchange between Lincoln and Weitzel took place (on April 3, 1865, some five months after Lincoln's reelection in 1864 but a few days before General Robert E. Lee's surrender at Appomattox).

Here is Graves's account of the exchange, which occurred soon after the fall of Richmond, Virginia:

As we came down the staircase [from CSA President Jefferson Davis's abandoned office at the Davis house], General Weitzel came in breathless haste, and at once President Lincoln's face lost its boyish expression, as he recalled that duty must be resumed. Soon afterwards Judge Campbell, General Anderson (Confederates), and others called and asked for an interview with the President. It was granted, and took place in the parlor, with closed doors. I accompanied President Lincoln and General Weitzel to Libby prison and Castle Thuner, and heard General Weitzel ask the President what he (General Weitzel) should do in regard to the conquered people. President Lincoln replied that he did not wish to give any orders on that subject, but, as he expressed it, "If I were in your place I'd let 'em up easy, let 'em up easy."

As both jwpat and Guy note, the expression "let [someone] up easy" refers to releasing someone with whom one has been fighting from a helpless position instead of exploiting the advantage to injure or punish the person further. Thus, P.N. Elrod, Dark Road Rising (2009), has this:

I pushed until his face was mashed against the metal and lifted his arm a few notches. Any more would break or dislocate it depending on where I put the pressure. He still struggled. "I've already been there, thanks to you and Hog Bristow."

At that name, and the emphasis I placed on it, he paused.

"We talk," I said quietly. "And maybe have a drink. You wanna get out of the cold?"

He thought it over, then nodded. I let him up easy, ready for another round. He rubbed his arm instead, his gaze sharp. "This is your club."

John Bartlett, Dictionary of Americanisms (1848) indicates that let up was in use in other fields by 1848, after originating as fight slang:

LET UP. A let up is a release ; a relief. An expression borrowed from pugilists.

[Example:] There was no let up in the stock market to day, and the differences paid on the maturing contracts were very large.—N.Y. Tribune

The expression "let [someone] up easy" isn't especially common in U.S. English today, but I think that most American English speakers would recognize the intended meaning in context, as in the quote from Elrod above.