What are the ideal pair of (everyday) elements to pair in a thermocouple
My book has a few metals/alloys listed in a Seebeck Thermoelectric Series:


  Antimony, Nichrome, Iron, Zinc, Copper, Gold, Silver, Lead, Aluminum, Mercury, Platinum-Rhodium, Platinum, Nickel, Constantan, Bismuth


There's also the Seebeck formula for emf of a thermocouple:

$$E_{\text{AB}} = a_{\text{AB}}\theta + \frac{1}{2}b_{\text{AB}}{\theta}^2$$

Here, $a_{\text{AB}}$ and $b_{\text{AB}}$ are constants that depend on the pair of metals $\text{A}$ and $\text{B}$ chosen. $\theta$ is the temperature difference between the junctions of the thermocouple. 

There's another defined quantity called thermoelectric Power, which is defined as $\frac{dE_{\text{AB}}}{d\theta}$.

The graph of $E_{\text{AB}}$ with respect to $\theta$ would be a parabola. If $a_{\text{AB}}$ and $b_{\text{AB}}$ are of opposite signs, then the parabola crosses the $x$-axis twice, and thus, the thermo-emf becomes zero for two values of $\theta$ and maximum for one value of $\theta$. 

From the above thermoelectric series, a iron-nickel might seem like a good option to consider building a thermocouple with, considering they're quite far away in the series and are easily accessible. For this couple, we get the values of thermoelectric constants as $a_{\ce{Fe-Ni}}=\pu{2.5 \mu V/°C}$ and $b_{\ce{Fe-Ni}}≈0$.Thus, you get:

$$E_{\ce{Fe-Ni}}=2.5\theta$$

If you're using your temperature range of 300°C-500°C for the hot junctions and room temperature (25°C) for the cold junction, you would get your thermo-emf in the range $\pu{0.687-1.1875mV}$.