They're not sought out in part becasue there isn't the huge deal of respect for them the way there would be toward the Founders or toward a great leader.
Of coruse, John Quincy Adams is easy if you want to say that you need to have one who is really well respected. Don't let him start drinking and you could have him serving in Congress for years longer. Even if he doesn't serve as long as Thurmond did he could be in his late 80s arguing for Brooks' expusion after he canes Sumner, and squawking about the traditions of liberty his father "helped to create and which were being destroyed by Southerners." He coudl be an interesting elder advisor to LIncoln as far as what to do about salvery.
But, by influential, are we talking about someone who really impacts things? Becasue Adams, while influential, might not make thigns much more different than they already were OTL; in fact, I can see him having the same midnset as Lincoln, slowly turning from wanting to eekp the Union together without touching salvery to gradual emancipation to the 13th Amendment becasue the South refuses to talk.
So, the most impactful might be Theodore Roosevelt. Living not only to be President in the 1920s but to have his son Quentin potentially as President later, that could really make thigns interesting.
Or, maybe Washington. True, he'd gotten out of politics, but that doesn't mean his voidce wouldn't be helpful. He and Jefferson were both feeling that slavery would die out; his voice added to Jefferson's could get Virginia to gradually emancipate salves starting in the 1820s, butterflying the Nat Turner rebellion (at least potentially and swinging things in the North's favor even more during a Civil War.