Nested definition: How can I define a function with a passed-in expression?

Try for example

SetAttributes[def, HoldAll]
def[s_Symbol, v_] := Function[Null, s[x_] := #, HoldFirst][v]

Unnamed functions just don't care :)

Other alternatives that should also work (but I would use the previous approach)

def[s_Symbol, v_] := Identity[SetDelayed][HoldPattern@s[x_], v];
def[s_Symbol, v_] := Unevaluated[s[x_] := "Hello"] /. "Hello" -> v

With your proposed definition style, the user of that function def has to know that v could/should/must depend on x for this to work; x really should be an argument of def. Perhaps something like this were better suited.

ClearAll[def]
ClearAll[f]
(*SetAttributes[def,HoldFirst]*)

def[s_Symbol, v_, vars_List] := 
 With[{h = s @@ (Pattern[#, Blank[]] & /@ vars)}, (h := v)]
def[f, x^2, {x}]

f[3]
(* 9 *)

I propose these:

SetAttributes[def, HoldAllComplete]

def[s_Symbol, v_] := SetDelayed @@ Hold[s[x_], v]

def[s_Symbol, v_] := With[{L := s[x_]}, L := v]

def[s_Symbol, v_] := Reverse @ Unevaluated[v := s[x_]]

The HoldAllComplete (or SequenceHold) attribute is necessary for an assignment such as:

def[q, Sequence[1, 2, x]]

head[q[5]]

head[1, 2, 5]

Also see:

• Enforcing correct variable bindings and avoiding renamings for conflicting variables in nested scoping constructs