What property of a BJT transistor makes it an amplifier?
A transistor on its own does not make it an amplifier.

The transistor needs a circuit around it to do the actual (signal) amplification.

Depending on the circuit a transistor can amplify current changes and/or voltage changes and that means power amplification. Power amplification means that you need a smaller power to control or output a larger power.

In my opinion, the most basic property of a transistor which results in (power) amplification is the current relation between base current \$I_B\$ and collector current \$I_C\$. Their ratio is often referred to as \$\beta\$:

$$\beta = \frac{I_C}{I_B} $$

This \$\beta\$ is also quite "visible" in the actual transistor as it is linked to the ratio between the doping levels of the emitter and the base. The emitter will have the highest doping level, the base has a lower doping level (it could be \$\beta\$ times lower) and the collector will have the lowest doping level.

So if we increase the doping level of the base region, \$\beta\$ will increase and "amplification" goes up.

Does that mean I will always get a higher amplification if I use a transistor with a higher \$\beta\$?

No, it depends on the circuit you're using.

In some circuits indeed a higher \$\beta\$ will give you more amplification.

For example, a transistor controlling a relay. When \$\beta\$ is increased, we could use a smaller base current.

In others it will not give you more amplification. 

For example, a Common Emitter amplifier, assuming we do not change the DC current \$I_C\$. In a CE amplifier, the voltage gain is \$gm*R_{load}\$. To get more gain we would need to increase \$gm\$ or \$R_{load}\$. Both can be done irrespective of \$\beta\$.