RandomInteger with equal number of 1 and -1

A combination of IntegerPartitions, RandomChoiceand RandomSample:

n = 30;
RandomSample @ RandomChoice @ IntegerPartitions[0, {n}, {-1, 0, 1}]

{-1, 1, 1, 1, 1, -1, -1, 0, -1, 1, 1, -1, -1, -1, 1, -1, 1, 1, -1, 0, 1, -1, -1, 1, -1, -1, 1, 1, -1, 1}

Total @ %

0

You can also do

RandomSample @
 PadRight[Flatten @ ConstantArray[{1, -1}, RandomChoice[Range[0, Floor[n/2]]]], n]

{-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, -1, -1, -1, 0, 0, -1, 0, 0, 0, 0, 1, 0}

Total @ %

0

For large n, the second approach is much faster than the first.

If you want to sample uniformly from all possible tuples that sum to 0, you can do the following:

zero[n_] := With[
    {
    ones = RandomChoice[
        Table[Multinomial[i,i,n-2i], {i,0,Floor[n/2]}] -> Range[0,Floor[n/2]]
    ]
    },

    RandomSample @ PadRight[
        Join[ConstantArray[1,ones],ConstantArray[-1,ones]],
        n
    ]
]

For example, here's a tally of the random 4-tuples summing to 0:

SeedRandom[1];
Tally @ Table[zero[4], 10^5]

{{{1, 1, -1, -1}, 5215}, {{-1, 1, 0, 0}, 5353}, {{-1, -1, 1, 1}, 5381}, {{1, -1, 0, 0}, 5167}, {{1, -1, -1, 1}, 5169}, {{0, 1, 0, -1}, 5189}, {{0, -1, 1, 0}, 5311}, {{-1, 1, -1, 1}, 5263}, {{0, -1, 0, 1}, 5376}, {{0, 0, 1, -1}, 5268}, {{1, 0, -1, 0}, 5303}, {{0, 0, 0, 0}, 5218}, {{1, 0, 0, -1}, 5220}, {{1, -1, 1, -1}, 5095}, {{0, 0, -1, 1}, 5245}, {{-1, 0, 0, 1}, 5313}, {{-1, 1, 1, -1}, 5253}, {{0, 1, -1, 0}, 5264}, {{-1, 0, 1, 0}, 5397}}

Looks pretty close to uniform sampling.

As a comparison, note the distribution using the accepted answer:

nonuniform[n_] := RandomSample @ PadRight[
    Flatten@ConstantArray[{1,-1},RandomChoice[Range[0,Floor[n/2]]]],
    n
]

Tally @ Table[nonuniform[4], 10^5]

{{{-1, 0, 1, 0}, 2739}, {{0, 0, 0, 0}, 33695}, {{0, 1, 0, -1}, 2771}, {{-1, -1, 1, 1}, 5682}, {{0, 0, 1, -1}, 2807}, {{1, -1, 0, 0}, 2697}, {{0, 0, -1, 1}, 2790}, {{-1, 0, 0, 1}, 2765}, {{-1, 1, 0, 0}, 2738}, {{0, -1, 0, 1}, 2654}, {{-1, 1, -1, 1}, 5482}, {{1, -1, -1, 1}, 5555}, {{0, -1, 1, 0}, 2768}, {{1, 0, -1, 0}, 2721}, {{-1, 1, 1, -1}, 5519}, {{1, 1, -1, -1}, 5512}, {{0, 1, -1, 0}, 2727}, {{1, 0, 0, -1}, 2710}, {{1, -1, 1, -1}, 5668}}

Not very uniform.

Given the lack of details in your specification, I suspect the following very simple approach will be adequate to your needs:

zeroSum[n_] := With[{
   n3 = Floor[n/3]
   },
  RandomSample@Catenate@{
     ConstantArray[-1, n3],
     ConstantArray[1, n3],
     ConstantArray[0, n - 2*n3]
     }]