f[x_] := x^2

DownValues[h] = DownValues[f] /. f -> h;
f[x_] := f[x] = h[x]

h is now equal to the original f, and f is a memoized version of it.

f[x_] := x^2

DownValues[h] = DownValues[f] /. f -> h;
f[x_] := f[x] = h[x]

h is now equal to the original f, and f is a memoized version of it.

Some kinds of definitions would require a more specific replacement rule than /. f -> h, for example recursive functions.