My dad purchased a lavalier microphone (one with a 3.5 mm connector) for business meetings and was unaware that it needed a power supply. Could a simple 3.5 mm to usb converter power it via USB? If not are there any cheap alternatives which could power it and allow it to plug directly into the computer?
It depends on the power expectations and circuit design of the microphone.
Some phone lavalier mics use a LR44 battery. Maybe yours does, too. The LR44 battery, for example, has a nominal voltage of 1.5V and 105 mAh. USB has 5V and 3A (USB-C). That's a lot more power than a device using a LR44 battery would expect. If you wired it up directly, you'd probably burn out the device. So, while USB is certainly capable to power the device, you'd need some circuitry in between to convert it to the power level that the device expects.
But, whether plugging in the 3.5mm jack into a USB->3.5mm female adapter will power the unit depends on the circuit design of the microphone.