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The top left part is a bridge rectifier that converts AC (DCC) signal into DC. You can solder to it's +/- (right hand side) a capacitor if it's small enough or a capacitor circuit ( resistor

But I'd also try to see the voltage on the empty "+ -" pads on the bottom. They seem to be designed for just this purpose, seemingly in parallel with an existing small size capacitor on the board. I'd power the board and measure the voltage on those pins.

Before the regulator

That board looks like it has a bridge rectifier and a voltage regulator (U1) on it, with some associated parts.  If you're going to add a capacitor, you'll get a lot more benefit out of adding it on the rectifier (input) side of the regulator than after it.  So tracing out the circuit to be sure where you're adding stuff would be useful.

There are pc traces on the surface of that board, they've just been painted over.  Even in that photo you can see hints of them in slight relief.  Shin a light across the board from a low angle from different directions and I bet you'll be able to trace the traces.  There may be similar painted over traces on the back of the board if you shine a low angle light on it.  That's such a simple circuit I very much doubt they sandwiched traces in the middle.

Half the circuits seem to be hidden

The top side of the board only seems to show half of the circuits. The back is totally smooth with no parts, solder points or sign of any traces. It appears that half of the traces are between the top and a back layer. Someone at the train club said they had seen boards with a similar arrangement to help prevent shorts.

The board seems to have two very tiny capacitors (C1 & C2). What are they for?

When I get home I'll try measuring the voltage on the + & - pads.

Recently did this

I recently tackled this exact question. First, the + and - pads are post regulator and are nominally a 4.4 v supply to the leds. The output lead of the regulator U1 is the one closest to the car center line of the PCB. It connects directly to the marked + pad. I cut this trace right next to the plastic lens over the leds, leaving a small pad immediately adjacent to the regulator output. This isolates the + pad such that it becomes the +v to the leds. I did this because I also wanted to insert a series resistor to knock down the brightness some so my cars don’t look like they belong in the subway. If you are good with the light level, you can just connect capacitance at the + and - pads and no need to cut any trace. 

i used a  https://www.digikey.com/product-detail/en/eaton-electronics-division/KR-5R5H474-R/283-2821-ND/1556249. This is 470MF at 5.5v. PLENTY! I think I would probably have been fine with the 100MF version of the same part (same package). It takes a good minute or more for the lights to come up to full brightness at power up, the regulator limits the charging current so no additional resistor/diode is needed. Once at fully charge, the lights are maintained for several minutes after power is removed. I bent and clipped the capacitor terminals as needed to prepare for attachment of wire leads in a manner that made sense for the intended location of the cap

Working on the output of the regular allows use of a 2 “cell” super cap, getting more capacity in less size since voltage rating for these is obtained by connecting individual units in series.

I connected the cap’s + terminal directly to the regulator output, ahead of the series resistor I added. (I chose 100 ohm, this is a subjective call as to what brightness level works for you. YMMV.) I connected the load side of that resistor to the pcb’s + pad to drive the leds. I had no difficulty slipping the cap into a non visible spot in the car but exact location and thus required lead length was different depending on the specifics of the car interior.

Chuck

Which side of the regulator

If you put the capacitor on the output side of the regulator (4.4 volts), the voltage across the LEDs will droop during dead spots as the capacitor discharges.  If you put the capacitor on the input side of the regulator, the capacitor could discharge about halfway (from 12 down to 6 volts) before you might begin to see the output voltage dip below 4.4.  It may be that the output side is easier and good enough to solve the problem.  But if the input side looks doable, consider it.

size v. capacity

Ken, in principle you are correct. Practically, in the installation as I described, I found no perceptible droop running across typical dead spots at slow speed once the cap had fully charged. The high capacitance possible in the small package at a lower voltage rating as well as the relatively small current draw in this application (further reduced by the additional series resistance) enable this. I mentioned in my initial post that a smaller value cap might work as well - it would charge quicker and discharge quicker and thus the droop would certainly be be apparent "sooner". Use of a smaller value standard electrolytic cap would be even more prone to this.

Of course in real life in the pre-HEP days, the storage batteries backing up axle generators didn't last forever when the train wasn't moving (but at least you didn't also lose heat in the winter when the lights went out! Assuming steam from the head end was maintained of course.)

I also found low cost and simplicity made my install in this situation more than good enough.