He would likely continue to be an important power broker in Republican Party politics for at least the next decade after 1869. There were no primaries at the time, the conventions decided everything. Lincoln was somewhat suspicious of big business (although in favor of policies that lead to the development of industry and he was a railroad lawyer) so he is likely to be part of what would become the progressive faction of the Republican Party. One could see him supporting Theodore Roosevelt's rise.
He would likely write his memoirs, which would probably be an American literary classic. Reading it (or at least passages) might be common in American High School American Literature classes.
If we believe that Lincoln was more successful in handling reconstruction (the usual ATL fantasy - civil rights for blacks is enforced, but his political adeptness restrains the worse abuses of the military governments. While not perfectly well treated, blacks in the South are able to retain their vote), then we may see some sort of continuing reconciliation role of Lincoln with the south along with supporting private organizations assisting the freedman adjust to emancipated life. He might make public appearances with various Confederates int he South and North to heal the wounds of division. And he might support development of institutions like the Tuskegee Institute that would provide continuing educationa nd skills for freed blacks. It would be much harder to roll back new rights for former slaves with Lincoln around trying to preserve his legacy. The two probably ensure some form of Republican Party survives in the South with an emphasis on individual homesteaders/farmers, steadfast unionists during the war, and former slaves. It means certain areas of the south where Republicans can win local elections, and perhaps certain southern states where Republicans might even win statewide elections.
I see him being against colonialism, and would likely speak out against the European scramble for Africa. If he survives long enough, he may not speak out against the Spanish-American War but he would likely be against the US holding onto the Phillipines afterwards and the annexation of Hawaii. He might be too old by then and might now have any effect other than to be quoted by people afterwards.