As insinuated by @Solar Mike's comment, a good first step is to think about the equation which defines elongation, Hooke's Law:
$$\begin{align}
\sigma &= \frac{F}{A} = E\epsilon \\
\therefore \epsilon &= \frac{F}{EA}
\end{align}$$
Now, you don't state this clearly, but I assume you're asking about a case of a rod under a constant force. Therefore, if we want to reduce the elongation, the only variables we have in hand are $E$ and $A$. If we increase either of these, the elongation decreases.
So, to decrease elongation, choose a material with a higher modulus of elasticity and/or increase its diameter.
As for your other suggestions:
- temperature: we usually talk about materials as having a constant modulus of elasticity, but that's actually a simplification. Most (all?) materials have an inverse relationship between temperature and modulus of elasticity (after all, over a certain temperature, the material simply melts, at which point it doesn't resist elongation all that much). So if you lower the temperature, you increase the modulus of elasticity. However, lowering the temperature also causes most materials to shrink, which decreases the cross-sectional area. So the effect of temperature changes will depend on the material (and will usually be minuscule for "normal" temperatures).
- hardness: this is a meaningless term in structural engineering. Do you mean the Mohr hardness scale (which ranks materials by what can scratch what)? This is irrelevant (though maybe correlated with Young's modulus, not sure). A more often used term is "stiffness", which is equal to $EI$ for bending and $EA$ for axial loads. We've already explored how this influences elongation.
- tensile and yield strength: these are irrelevant. Elongation is a function of the stress applied and the elastic modulus. Tensile and yield strength simply define maximum thresholds for the stress.
