Digging from http://www.befria.nu/elias/pi/binpi.html to get the binary value of pi (so that it was easier to convert into bytes rather than trying to use decimal digits) and then running it through ent I get the following for an analysis of the random distribution of the bytes:

Entropy = 7.954093 bits per byte.

Optimum compression would reduce the size of this 4096 byte file by 0
percent.

Chi square distribution for 4096 samples is 253.00, and randomly would
exceed this value 52.36 percent of the times.

Arithmetic mean value of data bytes is 126.6736 (127.5 = random).

Monte Carlo value for Pi is 3.120234604 (error 0.68 percent).

Serial
correlation coefficient is 0.028195 (totally uncorrelated = 0.0).

So yes, using pi for random data would give you fairly random data… realizing that it is well known random data.

From a comment above…

Depending on what you are doing, but I think you can use the decimals
of the square root of any prime number as a random number generator.
These should at least have evenly distributed digits. – Paxinum

So, I computed the square root of 2 in binary to undetake the same set of problems. Using Wolfram’s Iteration I wrote a simple perl script

#!/usr/bin/perl
use strict;
use Math::BigInt;

my $u = Math::BigInt->new("2");
my $v = Math::BigInt->new("0");
my $i = 0;

while(1) {
    my $unew;
    my $vnew;

    if($u->bcmp($v) != 1) { # $u <= $v
        $unew = $u->bmul(4);
        $vnew = $v->bmul(2);
    } else {
        $unew = ($u->bsub($v)->bsub(1))->bmul(4);
        $vnew = ($v->badd(2))->bmul(2);
    }   

    $v = $vnew;
    $u = $unew;

    #print $i,"  ",$v,"n";
    if($i++ > 10000) { last; }
}

open (BITS,"> bits.txt");
print BITS $v->as_bin();
close(BITS);

Running this for the first 10 matched A095804 so I was confident I had the sequence. The value v_n as when written in binary with the binary point placed after the first digit gives an approximation of the square root of 2.

Using ent against this binary data produces:

Entropy = 7.840501 bits per byte.

Optimum compression would reduce the size
of this 1251 byte file by 1 percent.

Chi square distribution for 1251 samples is 277.84, and randomly
would exceed this value 15.58 percent of the times.

Arithmetic mean value of data bytes is 130.0616 (127.5 = random).
Monte Carlo value for Pi is 3.153846154 (error 0.39 percent).
Serial correlation coefficient is -0.045767 (totally uncorrelated = 0.0).

Well, among other properties of a random number generator, you probably want it to be a normal number. And several answers in the math.SE question that inspired your question point out that currently pi is believed to be normal, but it has not been proven.

Such generator would be a pseudo number generator, i.e. given the same seed, the result would always be the same. This being said, in most frameworks, when you use the standard random number generator, there is the same issue of being pseudo-random.

The distribution of the digits seems to be quite similar to the standard random number generators¹, so the digits of π can be used for ordinary random number generation scenarios.

The issue is that the algorithm will probably be very slow, compared to the ordinary random number generators, so it’s not very useful in practice.

^{¹ I believe it’s true, but don’t have any proof. It would be interesting (and not to complicate) to do a comparison based on a large quantity of numbers.}

The randomness of digits of pi (or for that matter any other sequence) can be arguably tested by so-called ‘battery tests’. One popular battery test is George Marsaglia’s Diehard Battery Test. There is also NIST Special publication 800-22 that describes a number of such tests and the results of applying these tests to a number of physical constants, including – lo and behold – pi for over a million bits. The result of pi is given in Appendix B of the report and looks like this:

Statistical Test                            P-value
Frequency                                   0.578211
Block Frequency (m = 128)                   0.380615
Cusum-Forward                               0.628308
Cusum-Reverse                               0.663369
Runs                                        0.419268
Long Runs of Ones                           0.024390
Rank                                        0.083553
Spectral DFT                                0.010186
Non-overlapping Templates (m = 9, B = 000000001)          0.165757
Overlapping Templates (m = 9)               0.296897
Universal                                   0.669012
Approximate Entropy (m = 10)                0.361595
Random Excursions (x = +1)                  0.844143
Random Excursions Variant (x = -1)          0.760966
Linear Complexity (M = 500)                 0.255475
Serial (m = 16, 2m∇Ψ )                      0.143005

Is pi a good random sequence generator? Look at the above results (or search the meanings of the left column variable, if you have no idea what they mean), and check if it satisfies your need.