How to pick out the element in a list of lists which satisfies a particular condition?

A more mathematically intuitive formulation would be to use Intersection, as then the logic is self-explanatory. You can also create a custom container to make the sets orderless so that {1,2,3}, {3,1,2} and {2,3,1} are all treated the same (alternately, you can use Sort). For example:

equivalentQ[func_] := Block[{f},
    SetAttributes[f, Orderless];
    With[{set = # /. List -> f}, 
        Intersection[set, func@# /. List -> f] === set
    ]
] &; 

Sets that equal those transformed by func should yield True. With this, you can write your condition as

 cond = equivalentQ[Mod[# + 2, 4, 1] &]

(and easily extend it to other transformation functions).

1. Select:

   Select[list, cond]
   (* {{{1, 2, 3, 4}}, {{1, 2}, {3, 4}}, {{1, 3}, {2}, {4}}, 
       {{1, 4}, {2, 3}}, {{1}, {2, 4}, {3}}, {{1}, {2}, {3}, {4}}} *)
   

2. Cases:

   Cases[list, x_List /; cond[x]]
   

3. Pick:

   Pick[list, cond /@ list]
   

4. Reap & Sow:

   Last@Reap@Scan[If[cond[#], Sow[#]] &, list]
   

If my reading of your question is correct, this does the job:

sets = {{{1, 2, 3, 4}}, {{1, 2, 3}, {4}}, {{1, 2, 4}, {3}}, {{1, 2}, {3, 4}},
        {{1, 2}, {3}, {4}}, {{1, 3, 4}, {2}}, {{1, 3}, {2}, {4}},
        {{1, 4}, {2, 3}}, {{1}, {2, 3, 4}}, {{1}, {2, 3}, {4}},
        {{1, 4}, {2}, {3}}, {{1}, {2, 4}, {3}}, {{1}, {2}, {3, 4}},
        {{1}, {2}, {3}, {4}}};

Cases[sets, list_List /; Equal @@ Map[Sort, {list, Mod[list + 2, 4, 1]}, 2]]
   {{{1, 2, 3, 4}}, {{1, 2}, {3, 4}}, {{1, 3}, {2}, {4}},
    {{1, 4}, {2, 3}}, {{1}, {2, 4}, {3}}, {{1}, {2}, {3}, {4}}}

I failed to spell out a working solution for your problem the first time. Here is what I intended:

sortall = # /. List :> Composition[Sort, List]&;

Select[lis, sortall @ #  === sortall @ Mod[# + 2, 4, 1] &]

{{{1, 2, 3, 4}}, {{1, 2}, {3, 4}}, {{1, 3}, {2}, {4}},
 {{1, 4}, {2, 3}}, {{1}, {2, 4}, {3}}, {{1}, {2}, {3}, {4}}}

J.M.'s method is cleaner, and in this case without a downside that I can see. (The sortall function is useful if you want to sorts Lists and only Lists at any level in an expression.) I still prefer Select here:

Select[lis, Equal @@ Map[Sort, {#, Mod[# + 2, 4, 1]}, 2] &]