Electronic – using 1.8V, max 40mA to switch a 12V, 70mA load (MOSFET data sheets?)

You are looking for the Gate Threshold Voltage, marked Vgs(th) in the datasheet. There is a bit of a catch to watch out for here though.

For this particular MOSFET, if we look at the relevant bit of the datasheet:

We can see it's given as a minimum of 0.6V, and a maximum of 1.2V. This looks promising for your 1.8V logic output. However, the catch is that this is for a drain current of only 250uA, so it's not of much practical use, since we'll need more than that to pull the PMOS gate low with the 4.7kΩ resistor, we need to make sure the drain current is acceptable at 1.8V.
To get a better idea of what voltage we need at the gate to sink useful currents at the drain, we would usually need to take a look at the graphs for Id over Vgs and Id over Vds for different gate voltages is also useful:

Unfortunately both only goes as low as a gate voltage of 2.25V, so we don't really know how much current the drain will sink at 1.8V. Since at 2.25V Id is over 10A, it's a reasonably safe bet that at 1.8V we'll be fine sinking 3.3V / 4.7kΩ = 0.7mA, but we can't be certain, which is ideally what we need to be.

Solution options:

• Use a different N-ch MOSFET with a guaranteed drain current of over 0.7mA at 1.8V Vgs given in the datasheet.
• Use any small NPN transistor which has a Vbe of ~0.7V and will be turned on easily with 1.8V (with current limiting base resistor of course. For a gain of e.g. 100, assuming you want to pull as low as possible, calculate using (1.8V - 0.7V) / (0.7 / 100) = ~150kΩ, then halve this to compensate for gain reduction at saturation, e.g. 75kΩ or less. Another conservative rule of thumb is to assume a gain of 20-30)
• Do your own tests with a few IRLML2502s and confirm there is more than enough drain current at a Vgs of 1.8V.
• Increase R2, so less current is needed to pull the gate down.

Personally I'd go for using an NPN, since it's cheaper - you could get a suitable part for a couple of pence, compared to the 33 pence for the MOSFET.
For sticking with this part though, unless you need fast switching, or there is a lot of noise present, I'd probably go for simply increasing R2, to around 15kΩ, so you only have to sink 3.3V / 20kΩ = 220uA. This is less than the value given at a max of 1.2Vgs, so you can be certain it will be able to pull the gate down easily.

One other possible option which I use regularly is to use an IO in open drain mode - many microcontrollers have certain pins which are tolerant of higher voltages than their supply voltage, so if your micro has a 3.3V tolerant pin which can be used in open drain mode (you don't give the part number so I can't check this, although you may have done so already) then you can do away with Q1 completely and use this instead.

What you build is a source follower which acts sort of like a voltage copier, it copies the gate voltage to the source (with a Vgs drop in voltage).

What is needed for Rds to be as low as you want it is that Vgs is maximised so the source needs to be grounded.

Alternatively you can use a PMOS if the grounds need to be common. Do note that the polarity at the gate of the PMOS switch is reversed compared to an NMOS ! Note also that the PMOS circuit is the same as the middle NMOS circuit but upside-down !

^{simulate this circuit – Schematic created using CircuitLab}