how to design a switch which works when an input of 0.10 is applied
0.15V won't drive the opto's input, since it's input LED has a typical forward voltage of 1.25V.

If 0.15V is really all you have access to, then you will need either a comparator or opamp with input range that includes it's lower supply rail.

Basically, you set the inverting input (-) to e.g. 75mV, then attach the signal to the positive input. Something a bit like this, but with the divider set for 75mV (e.g. 49.25kOmega; and 750&Omega; - using a pot to trim would make it easier to set):



If your power rail is prone to fluctuate, a voltage reference IC for the divider supply would be better.

If the output signal you have is e.g. 5V when the alarm is off, then you could do similar to the above but swap the inputs and have your reference connected to the + input at e.g. 2.5V. This would give you a lot more tolerance to supply variation or noise.  

If it turns out to be an AC signal, then you would need to add a simple peak detect circuit in front of your comparator. You will suffer a drop in your output voltage, so given the small voltage to start with, a Schottky diode for D1 would be best:



Simulation - blue trace is 150mV 1kHz square wave switched on for 500ms at 100ms, the green trace is the output voltage at the top of R1/C1:



EDIT

Okay, you say your 150mV output is DC, which makes things a bit simpler. In your diagram you are using a MCT2E Optocoupler, which has a maximum input photodiode current spec of 60mA, and ~20mA typical in order to turn the phototransistor fully on. Since the phototransistor is only driving the base of the relay drive transistor in your circuit, we probably don't need it fully on, but unless you are trying to conserve power, we'll set things up for a 20mA drive to make things simple.
Note that in your circuit you have two layers of isolation - one from the opto and one from the relay, but that's fine if this is how you wish to do things.

Okay, you need a comparator with an input range that includes ground. There are lots of options of you check on places like Farnell, etc. We are using LTSpice to simulate, so let's pick one from LT - the LT1018 is a random choice in the LTSpice library which has rail to rail out and it's input includes ground - any similar comparator with an input which ranges to ground, and can drive the 20mA would do. This one can only sink current, so we need a pullup resistor on the output.
For the input, we need to set the reference voltage at the non-inverting input (+) to somewhere between 0 and 150mV - let's pick ~75mV. As long as it's far enough away from either 0 or 150mV you should be fine.
For the output, we need to limit the current to 20mA. To do this, we take a look at the typical photodiode forward voltage, note out supply voltage and calculate using this formula:

Rout = (Vsupply - Diode_Vf) / I_diode = (5V - 1.25V) / 20mA = 187.5&Omega; 180&Omega;, 200&Omega; or 220&Omega; is near enough.

So we end up with a circuit like this:



Simulation:



We can see in the top plot the threshold voltage and the input voltage from the buzzer. We can also see the hysteresis provided by R4, which helps to prevent glitches on switchover by changing the positive going and negative going thresholds (see here and here for some explanation)