If you have another, working, Uno, and both have socketed microcontroller chips, you could use the known board to program a test sketch onto the unknown board's chip, put it back on the unknown board, and run it.

I like the following (pseudo-coded) test sketch for quick-checking my hand-built boards; it's just as handy for testing an unknown factory-built one. It's not an exhaustive test, it just checks for basic digital I/O, serial I/O, and shorts or opens:

forever,
  for each digital I/O pin:
    write high;
    delay 100ms;
    write low;
  end;

  write "Hello, World!\n" to serial output;
  while serial character available,
    write character to serial output;
  end;
  write "\n" to the serial output;
end;

• A test LED - an LED with its resistor soldered to it - connected between any I/O and ground should blink every 2 seconds. If it doesn't, that pin may have failed or have a bad connection.

• Two test LEDs connected to numerically adjacent pins (they may not be physically next to one another) should blink in sequence. If they blink together, it indicates a short, probably a sholder bridge, between them.

• The terminal should say "Hello World!" every two seconds. If not, that pin or the FTDI cable or on-board USB/Serial chip may have failed.

• Type something on the keyboard and it should be repeated back to the serial terminal (if the previous test passed). If not, again that pin or the FTDI cable or on-board USB/Serial chip may have failed.

If you have another, working, Uno, and both have socketed microcontroller chips, you could use the known board to program a test sketch onto the unknown board's chip, put it back on the unknown board, and run it.

I like the following (pseudo-coded) test sketch for quick-checking my hand-built boards; it's just as handy for testing an unknown factory-built one. It's not an exhaustive test, it just checks for basic digital I/O, serial I/O, and shorts or opens:

forever,
  for each digital I/O pin:
    write high;
    delay 100ms;
    write low;
  end;

  write "Hello, World!\n" to serial output;
  while serial character available,
    write character to serial output;
  end;
  write "\n" to the serial output;
end;

• A test LED - an LED with its resistor soldered to it - connected between any I/O and ground should blink every 2 seconds. If it doesn't, that pin may have failed or have a bad connection.

• Two test LEDs connected to numerically adjacent pins (they may not be physically next to one another) should blink in sequence. If they blink together, it indicates a short, probably a solder bridge, between them.

• The terminal should say "Hello World!" every two seconds. If not, that pin or the FTDI cable or on-board USB/Serial chip may have failed.

• Type something on the keyboard and it should be repeated back to the serial terminal (if the previous test passed). If not, again that pin or the FTDI cable or on-board USB/Serial chip may have failed.