Voltage Drop
Well, if you put 5V into your barrel jack, your UNO's onboard 5V regulator is going to drop that down to 3.5V or so. You must have more than 5V into the regulator to get 5V out of it. This is referred to as the regulator's "drop out" voltage.
The specs for an UNO say to power it from at least 7V for that reason.
As an experiment you can power it this way without your lights attached and measure the "5V" header to GND to see this happening.
Power Dissipation
Linear regulators like that found on the UNO are not efficient. They waste energy as heat in proportion to the current through it and voltage dropped by it.
If you provide it with 7V at the regulator and your project draws 0.5 amps from the regulator, you will try to generate 0.5A * (7V - 5V), that is 1 Watt of heat. Your regulator can probably dissipate half that with how it's mounted on the UNO and normal ambient air temperature.
Note this situation gets worse if you put 12V (for example) into the barrel jack, as you'd be dropping 7V across the regulator.
Many linear regulators have thermal overload protection built into them. They'll stop producing as much voltage at the output as they heat up to try to survive the event. And that's likely what you're seeing if you're trying to power your christmas lights from the regulator.
If you want to see this yourself might try very briefly tapping the regulator with your finger; it may be blazing hot. You might just dampen your finger and hold it near the regulator's surface to feel the heat coming off of it.
You will more than likely need to route power around the Arduino board to your lights to prevent this.
