This will eventually stop…

C#, 94 85 bytes

My first answer!

using System;s=>{var r=new Random();for(var i=2;r.Next(i++)>0;)Console.Write(s+" ");}

Previous attempt (I liked that goto):

using System;s=>{var i=2;var r=new Random();a:if(r.Next(i++)>0){Console.Write(s+" ");goto a;}}

Ungolfed:

using System;
class P
{
    static void Main()
    {
        Action<string> f = s =>
        {
            var r = new Random();
            for (var i = 2; r.Next(i++) > 0;) Console.Write(s + " ");
        };

        f("test");

        Console.ReadKey();
    }
}

Note: in C# the Random.Next(N) method returns a nonnegative integer in the [0, N-1] range, so we can just check that the number returned is greater than 0.

Pyth, 7 bytes

WOh=hZQ

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How it works

Pseudocode:

while rand_int_below(1 + (Z += 1)):
    print(input)

R, 47 46 43 bytes

43 bytes due to Robin Ryder in the comments.

s=scan(,"")
while(sample(T<-T+1)-1)print(s)

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Explanation

s=scan(,"")  # Takes input from stdin.
             T<-T+1    # T is 1 by default, so this
                       # evaluates to 2, and will increment
                       # at each step.
      sample(T<-T+1)   # Take a sample of size 2, i.e. generate
                       # a list of integers from 1 to 2 in random order
      sample(T<-T+1)-1 # Subtract one from every element of this list.
while(sample(T<-T+1)-1)# while() will treat the first value in this list
                       # as a logical value, i.e. FALSE for zero and TRUE
                       # for nonzero values. The other elements of the list
                       # are ignored, triggering a warning.
                       print(s) # print s