How to replace numbers by letters in a special way?

lists.{"A", "B", "C"} /. 0 -> "1" /. Plus -> StringJoin 

{"1", "C", "B", "BC", "A", "AC", "AB", "ABC"}

% // ToExpression

{1, C, B, BC, A, AC, AB, ABC}

One possibility is to use ReplaceAll (the short form of which is /.) with a list of rules. There are probably more concise ways of doing this that I'm not seeing.

{{0, 0, 0}, {0, 0, 1}, {0, 1, 0}, {0, 1, 1}, {1, 0, 0}, {1, 0, 1}, {1, 1, 0}, {1, 1, 1}} /. 
{{0, 0, 0} -> 1, {0, 0, 1} -> C, {0, 1, 0} -> B, {0, 1, 1} -> BC, 
 {1, 0, 0} -> A, {1, 0, 1} -> AC, {1, 1, 0} -> AB, {1, 1, 1} -> ABC}
(* {1,C,B,BC,A,AC,AB,ABC} *)

However, I would recommend surrounding each of those capital letters with quotation marks to make them strings, or else using lower case letters. All built-in functions in Mathematica begin with capital letters, and there are a number of single character capitals reserved by the system such as C, D, E, I, K, N, and O.

Here's a way to do it (it's bad practice to put an involved answer in a comment so I'm moving it here):

lists = {{0, 0, 0}, {0, 0, 1}, {0, 1, 0}, {0, 1, 1}, {1, 0, 0}, {1, 0, 1}, {1,
     1, 0}, {1, 1, 1}};
With[{a = ToUpperCase@Alphabet[]},
 StringJoin /@ MapIndexed[If[# == 1, a[[#2[[-1]]]], ""] &, lists, {2}] /. 
  "" -> "1"
 ]

(* Out: {"1", "C", "B", "BC", "A", "AC", "AB", "ABC"} *)

Basically I directly translated what you said into code.

Here's a fun way that involves using the 0/1 as a position, extracting that, then doing the string join:

With[{a = Prepend[ToUpperCase@Alphabet[], ""], sublen = Length@lists[[1]]},
 Replace[
  StringJoin /@
   Partition[
    a[[
      Flatten[1 + ConstantArray[Range[sublen], Length@lists]*lists]
      ]],
    sublen
    ],
  "" -> "1",
  1
  ]
 ]

Some tests show that it might be minimally faster than the former