How do you create a random string that's suitable for a session ID in PostgreSQL?

You can fix your initial attempt like this:

SELECT md5(random()::text);

Much simpler than some of the other suggestions. :-)

I'd suggest this simple solution:

This is a quite simple function that returns a random string of the given length:

Create or replace function random_string(length integer) returns text as
$$
declare
  chars text[] := '{0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z,a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z}';
  result text := '';
  i integer := 0;
begin
  if length < 0 then
    raise exception 'Given length cannot be less than 0';
  end if;
  for i in 1..length loop
    result := result || chars[1+random()*(array_length(chars, 1)-1)];
  end loop;
  return result;
end;
$$ language plpgsql;

And the usage:

select random_string(15);

Example output:

select random_string(15) from generate_series(1,15);

  random_string
-----------------
 5emZKMYUB9C2vT6
 3i4JfnKraWduR0J
 R5xEfIZEllNynJR
 tMAxfql0iMWMIxM
 aPSYd7pDLcyibl2
 3fPDd54P5llb84Z
 VeywDb53oQfn9GZ
 BJGaXtfaIkN4NV8
 w1mvxzX33NTiBby
 knI1Opt4QDonHCJ
 P9KC5IBcLE0owBQ
 vvEEwc4qfV4VJLg
 ckpwwuG8YbMYQJi
 rFf6TchXTO3XsLs
 axdQvaLBitm6SDP
(15 rows)

Building on Marcin's solution, you could do this to use an arbitrary alphabet (in this case, all 62 ASCII alphanumeric characters):

SELECT array_to_string(array 
       ( 
              select substr('abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ0123456789', trunc(random() * 62)::integer + 1, 1)
              FROM   generate_series(1, 12)), '');