Without his disasterous Presidency he'll be remembered as the man who fought for the Union despite everyone in his home state loathing him and dying for it.
Well, not everyone. Almost half of Tennessee was solidly pro-Union.
In any case, this POD is really about Lincoln remaining President and overseeing reconstruction. The legacy reputation of Andrew Johnson being of inconsequential importance except as its use to bind Unionist Southerners closer to the Republican Party.
As such, we'll probably see a more successful Reconstruction within limits. A better supported freedman class that remains politically important combined with Southern Unionists and Northern immigrants, may be able to hold a few states of the South for the Republicans long term. Grant is still likely to become President after Lincoln, but he'll inherit a situation where it'll be much harder for "redemptionists" to overturn the civil rights of freedmen.
The Republican Party won't be as bitterly divided between radicals and moderates since Lincoln is a natural bridge between the two. The country also won't be exhausted about a Washington that doesn't work since there is no impeachment crisis. Popular support in the North will be much higher to keep Reconstruction going and successful.
Eventually, the troops will need to be withdrawn and some of the Southern states will regress as IOTL, but I think there is a real chance for several of the Confederate states to retain civil protections for blacks and competitive elections where Republicans can win.
If any of the Southern states goes for Hayes in 1876, there is no constitutional crisis or commission ruling. A Republican administration may be more willing to continue its support of freedmen. However, after 20 years of Republican rule, then Hancock could very well win the 1880 election for the Democrats. It would be interesting to see if this would actually play out, and what that may mean for freedmen and their families if the first Democrats in office is an out-and-out Civil War hero.
Like all reconstruction scenarios though, many questions as to its true success.