The Republicans by not making a firm choice in that election basically let the choice be made for them by events.
It illustrated the necessity for big city Dems to embrace civil rights to keep their machines in power. Northern Democrats became less timid and with Democrats in control on the hill, it meant that the Democrats would be the ones in the driver's seat the next time the won the Presidency.
That is exactly what ended up happening in the 1960's. Disappointed with the black vote result in 1956 (which was ironically partially due to Republican foot dragging) led to a feedback loop where Republicans would put in less effort each time and use that to justify even less the next time. Meanwhile the South kept trending Republican, so in the 1960's the Republican strategy was geared towards getting the middle class Southern whites to solidify as Republicans and poorer Southern whites to default to the Republicans as the "less pro-civil Rights party".
Leaving the more direct 1964 approach aside, in the late 1960's, there was a general desire on the part of Republicans to increase black voting, not because they were necessarily eager to right this century old wrong, but to basically inject Democratic primaries with majority black voting demographics and thus scare the racist whites into becoming Republicans, nor less than Kevin Phillips himself advocated this approach. A similar impetus was likely the motivating factor behind the more aggressive implementation of the VRA in redistricting in the early 1990's, because that would further polarize southern politics based on race.
