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VertexReplace

Renames the vertices of a graph, in its vertex list and in every edge.

$ wo 'VertexList[VertexReplace[Graph[{1 <-> 2}], 1 -> 5]]'
{5, 2}

$ wo 'ToString[EdgeList[VertexReplace[Graph[{1 -> 2}], 2 -> y]], InputForm]'
{DirectedEdge[1, y]}

Renaming onto a vertex that is already there merges the two:

$ wo 'VertexList[VertexReplace[Graph[{1 <-> 2, 2 <-> 3}], {1 -> 3}]]'
{3, 2}