The second version is in theory more efficient, decompiling to:
public static java.lang.String function1(int);
Code:
   0: ldc           #2                  // String
   2: astore_1
   3: iload_0
   4: bipush        10
   6: if_icmple     12
   9: ldc           #3                  // String a
  11: areturn
  12: ldc           #4                  // String b
  14: areturn
whereas the version with the assignment decompiles to:
public static java.lang.String function1(int);
Code:
   0: ldc           #2                  // String
   2: astore_1
   3: iload_0
   4: bipush        10
   6: if_icmple     15
   9: ldc           #3                  // String a
  11: astore_1
  12: goto          18
  15: ldc           #4                  // String b
  17: astore_1
  18: aload_1
  19: areturn
where it can be seen that the additional variable is created and returned.
However in practise the difference in actual runtime performance should be negligible.   The JIT compiler would (hopefully) optimise away the useless variable, and in any case unless the code was in a hot code path according to your profiler then this would certainly count as premature optimisation.