Is it possible to Clear (recover memory from) a specific index to a variable, while leaving other indices to the same variable untouched?

Byte count is 64, for entry in indexed variable. Even if there is no data

As mentioned in comment, use =. to clear specific index

Too long for a comment:

ByteCount[mytest[data]] measures the memory required for the expression mytest[data]. The ByteCount of a Symbol is always zero; see the docs for ByteCount:

Symbols are effectively always shared, so they give 0 byte count:

In other words (I think), it does not count the memory required to store the symbol in the symbol/hash table.

The OP's first ByteCount[data] computes ByteCount[{1, 2, 3}] because data evaluates to the expression {1, 2, 3} before ByteCount is called. Consider

Trace[ByteCount@data]
(*  {{data, {1, 2, 3}},
     ByteCount[{1,2,3}],
     112}  *)

Also consider the difference between the evaluated and unevaluated data:

data = {1, 2, 3}; 
ByteCount /@ {data, Unevaluated@data}
(*  {112, 0}  *)

@Nasser's example ByteCount[a[1]] gives 64 bytes and the OP's ByteCount[mytest[data]] gives 48 bytes (after data is cleared). Here is an accounting of it:

Clear[a, b];
ByteCount@ 1
ByteCount@ a[b]
ByteCount@ a[1]
(*
  16  - storage of one integer
  48  - storage of the head--argument tree expression
  64  - total
*)

One can carry this accounting game further. For instance, ByteCount@ 2[1] gives 80. However, beware that according to the docs,

ByteCount will...often give an overestimate of the amount of memory...needed....

Back to the principal question, which has already been answered, how to free up memory when a value is no longer needed:

• Unset will remove a single value (definition).
• Clear will remove all values (definitions) but not attributes, options, defaults or messages.
• ClearAll will remove all values, attributes, options, defaults and messages.
• Remove removes the symbol (thereby all things associated with it) from the symbol table.

See the docs for more information. As has been noted in the comments and @Nasser's answer, Unset is the desired solution in the OP's case.