So I want to make a Lazy Susan that is rotating constantly in one direction. I also want to have some other things on top such as neon signs, moving things, and etc. Problem is I want this to all run off of one DC source. Problem is I cannot figure out how I wire it so that things don't get tangled since the entire thing is constantly in motion.
You are looking for a slip ring. This is a device designed to do exactly what you describe - transmit power and/or signals to a rotating object. Generally, they work by having a rotating circular contact on one side and a spring-loaded pin which pushes against it on the other.
They are not super reliable long term or in harsh conditions, but should be fine for your project. More expensive ones are generally better and last longer. The other alternative is inductive power transfer/signalling, but this will be harder to get right first time, so might not be ideal for a hobby project.
An alternative to slip rings is the rotary transformer. This consists of two cup-shaped cores that face each other, with the windings inside the cups. If you drive the primary at a high frequency (we used 25 kHz), the cups can be ferrite and the whole thing can be quite compact (we did about 100W in a unit that was about 1" thick overall and about 3" in diameter).
How about using two rings of ball bearings and use each ring as the contact. No need to add pins and contacts...
Just to add, after taking note of the comments, that I was thinking of a low voltage supply here 5v or 12v just for leds or somesuch - I was not intending to suggest a 110VAC or 230VAC solution.
The other option that you may not have considered is to put all of your power consuming items, including the motor, and the batteries or other power source onto the lazy Susan. No electrical connectors to the outside world required then.
It has occurred to me that if you are driving the lazy Susan at a reasonable rate you are also in a situation where you can generate power on it.
If either on the central spindle or round the edges you could mount fixed magnets you could use appropriately placed coils to generate power on the moving platform.
If you were looking for really low tech you could even use an old push bike dynamo on the rotating platform running against a stationary surround of some sort.
Slot cars get their power from braid running on conductive tracks. You might be able to glue flattened coax cable shield (braid) in two concentric circles on your base board and then use braid "brushes" to pick up the voltage.
Or use one circle and brush and pick up the other side of the voltage via the central pivot.
Or use the central pivot as one contact, run the braid around the outside of the platter and use a horizontal brush.
Or think about the good old Dodgem cars where the floor was one contact and the wire mesh ceiling the other. Cover the whole of the base board with heavy duty domestic aluminium foil (leave a hole in the centre). Attach a brush to anywhere on the platter as one contact and use the central pivot as the other contact.
I looked into this recently building a zoetrope with LCDs and a microcontroller on the spinning top. I ended up making a DIY slip ring assembly from copper sheet and some carbon brushes designed for drill motors. It worked out surprisingly well (still good after 100,000 or so revolutions).
Passing the drive shaft through the transmitter and attaching the receiver to the underside of the lazy Susan worked pretty well if you can keep them close together. It didn't provide enough current for my application (I needed about an amp and these give about 500mA comfortably).
For my application I also needed to pass very low-rate uni-directional data from the base to the top, rather than add to my slip ring assy, I used IR LEDs and receivers, that worked really well.
Nicola Tesla said you could have a toothed ferromagnetic wheel on the rotor and stator each with windings. A combination transformer and synchronous clock motor. That avoids the legal expenses that you will have with liquid mercury, ozone poisoning, or electrocution.
Tweed's rotary transformer is probably the best solution when it is possible to wrap a coil around the axis and have a ferromagnetic core for the axel. But you could have a set of several smaller coils around the edge with similar coils coupling to them on the rotating platform. The power from each would be intermittent, but if they were properly spaced, you might meet your needs. Do expect nasty 60 Hz vibrations as the coils pass each other... unless you drive them at 100 kHz and use lightweight ferrite cores... If you want to go to high frequency, you might do some kind of capacitive coupling, as suggested by that great Scottish poet, R.F.Burns.
Use two rings, as with the ball bearings, but eliminate all wear and arcing with rings of liquid mercury or other conductive fluid. Don't drink it! Of course air is a conductive fluid at high voltage, and the display would earn respect of your guests and discourage nibbling of the marzipan and aboutargo on the rotating platform.