Electrical – IC 558 Quad timer chip, single timer in monostable mode

You really should post a schematic instead of all the hand waving. However, there is one obvious flaw in your description. One side of the relay could should be connected to the collector of the transistor as you say, but the other side must be to the power supply, not ground.

When the transistor is off, the collector is effectively not connected or left floating. When the transistor is turned on, it will pull the collector low. Since that end of the coil will be low, the other end must be high for there to be voltage accross the coil so that the relay can turn on.

Here is how you should hook things up:

Don't leave off D1. It is there to catch the inductive kickback from the coil when it is suddenly shut off. Without it, the coil will generate high voltage and fry the transistor.

You can't use the main timing resistor and capacitor to generate your trigger signal. As you discovered, this is exactly equivalent to the astable configuration.

Instead, you need to provide a separate resistor (to Vcc) and capacitor (to ground) for the Trigger pin. The time constant for this pair should be short relative to the main timing period, but long relative to the risetime of the power supply.

These components will hold the trigger low while the chip powers up, but then allow it to go high a short time later. Once the timer has timed out, the main timing capacitor will be discharged, but the trigger capacitor will not.