As I remember the War on Poverty, it started with people (like Robert Kennedy and LBJ) paying attention to West Virginia and Mississippi, finding examples of extreme hunger among the rural poor and the elderly. Then there was a famous book on the subject, was it called Poverty in America--no, it was "The Other America" by Michael Harrington.
So LBJ picked up the War on Poverty as part of his Great Society. Medicaid and Medicare were part of it, probably the most enduring part, but food stamps, which had been revived as a pilot program under Kennedy also got expanded.
LBJ famously said that the Civil Rights Aid had delivered the South to the Republicans. (A quote which may be too good to be true* but certainly represents reasonable fears at the time.)
What I want to note is that the results of the War on Poverty, plus other factors, like spending money on defense and space, the expansion of air conditioning, etc. have been good for the South. This map shows that the biggest share of improvements in poverty since 1960 has been in the South. Perhaps most important for the South has been the reduction of the civil rights issue: race relations are no longer a Southern problem, they're a national problem.
* From a footnote to this analysis: "
Most accounts, particularly by Bill Moyers,
claim that the president made this statement to him following the signing of the Civil
Rights Act. Lady Bird Johnson and Harry McPherson trace the statement to a conversation
after the signing of the Voting Rights Act in 1965. See Bill Moyers, “Second Thoughts:
Reflections on the Great Society,” New Perspectives Quarterly 4 (Winter 1987);
Jan Jarboe, “Lady Bird Looks Back: In Her Own Words, A Texas Icon Reflects on the
Lessons of a Lifetime,” Texas Monthly (December 1994): 117; “Achilles in the
White House: A Discussion with Harry McPherson and Jack Valenti,” Wilson
Quarterly 24 (Spring 2000): 92.