First an apology.... this could be a long post!
The mystery part will remain a much of a mystery unless the exact location in the circuit can be identified. That said, an educated guess can be made based on the information so far. It is almost certainly there to protect something from a voltage spike. If it goes to the ECU, my educated guess would be that it was added to the design of the Lag due to problems experienced with a previous design. I have seen threads where the delicate French electronics can be damaged by replacing the battery, so this would make sense. (as previously stated, some like to keep the sidelights on when changing the battery as this helps stop and spike when the battery is reconnected)
Now some other answers...
When capacitors fail, they will normally go 'open circuit' or 'short circuit'. When they go 'open circuit' it has the same effect as removing them from the circuit. When they go 'short circuit' it has the same effect as replacing it with a piece of wire. The effects of either of these failures depends on how they are used in the circuit.
If we assume that the mystery part is spike suppression, going 'open circuit' will simply mean that whatever it is protecting will be vulnerable to spikes, so you probably won't notice any symptoms of failure. Going 'short circuit' will effectively connect whatever it is connected to down to earth. If it is a 12V supply, then it would either blow a fuse, or whatever it is connected to would not work as it would not get 12v where it needs it. In this case you would almost certainly notice failure symptoms.
Now on to 'memory'....
An EPROM is an IC (integrated circuit) or 'chip' which is Erasable Programmable Read-Only Memory. These are used for holding information used that is not changed. There is a window on the top of the IC that is used for erasing. The IC is removed from the PCB (printed circuit board) and put into an EPROM eraser with is a box with a UV light. The UV light will erase the information on the EPROM. EPROMs are programmed by copying the required information from one device (could be another pre-programmed EPROM or a PC) onto the erased EPROM.
A further development from the EPROM is the EEPROM. This is Electronically Erasable Programmable Read-Only Memory. They are identical in use to EPROMS, but can be erased and programmed electronically, so there is no need to expose them to UV light.
I could also start to talk about other programmable devices like PLC (Programmable Logic Controller) but if you have managed to get this far into the post without falling asleep, going further will send you into a coma!
Bottom line - ANY device that is progammable will need programming. If you buy a programmable IC using the type number of the device it will be blank and of no use unless you can program it.
The good news is that it is highly unlikely that a 'goosed EPROM' is causing your fault.
OMG this post is getting epic!!!
Right.... Remember devices like EPROMs, EEPROMs, PLCs, Flash ROM and the like are used for storing information that does not change, like the engine mapping. If you have the ECU re-mapped, then they will change this information to increase performance and the like. If this is held on an EPROM, they must remove the EPROM, erase it and reprogram it with new information. If this information is held in something like a Flash ROM, then re-programming can be performed with the right kit (PC and interface) without removing the ECU from the car.
All of the above are known as 'non volatile'. This means that once programmed they do not need any power to retain their memory.
So far, we have only talked about information that is not changed. If we look at information that is routinely changed (trip computer information, odometer reading etc) then we use a different type of memory called RAM or Random Access Memory. ROM is exactly what is says... Read Only Memory. It can be read from, but not written to. RAM can be read from and written to.
The problem with most RAM is that it is volatile, meaning that when power is removed from the IC, all the information is lost. This means that you need something to keep it alive. This can be achieved in a number of ways:
1) Always keep power applied to the device. Anything on a car that is lost as soon as you remove the battery will use this method.
2) A capacitor (keeps alive for minutes). If you look at the label in the battery compartment of your SKY remote, you will see that it says 'insert new batteries within 10 minutes of removing the old ones' This is because if you programme it to work your TV, this information will be kept alive using a capacitor, and will be lost after a few minutes.
3) Rechargeable battery, or Lithium battery (keeps alive for weeks, months or even years). An example would be the BIOS battery in you PC. You could turn your PC off for a long time and when you boot it back up your changeable BIOS settings (like time and date) are still there. When these batteries eventually fail (as they will) you get a message saying something like 'time and date not set' when you boot your PC because the information was lost.
Also available is Non Volatile RAM or 'NVR' this means that when power is removed, the information is retained, so no stay alive is required.
Surely your not still reading this are you??
The odometer is very important and not worth risking any of the previously mentioned stay alive methods, so I would have thought it would use NVR.
To figure out what is going on with your display, it would be useful to know what the normal behaviour is when you disconnect the battery. The trip computer, clock etc are unimportant and I would be surprised if they have anything to keep the settings when the battery is disconnected. Having said that, if someone can confirm that these settings are retained when the battery is disconnected, then I would be looking for whatever is keeping them alive when the power is disconnected, as I know from experience that failing memory backup batteries can lead to very strange symptoms.
The fact that your display works (eventually) would first point me towards a poor connection. Unfortunately this could be anything from a poor connection in a connector or a poor earth connection right through to a hairline crack in a track on a multi layer printed circuit board that will be impossible to find.
Ideally you would borrow a known working display, try it and at least isolate the fault to the display or the connections to it. Without knowing this I think you could be spending a long time trying to find the fault!
Earth problems are very common. Can you identify an earth connection for the display? If you can, I would make sure that it was a good 'solid' connection.
Final notes..... although I am an electronic engineer by training, I am no expert on automotive electronics, so I apologise if any of the above is not applicable in automotive electronic design. Also, it is getting late and this is an EPIC post, so apologies for any spelling mistakes or general ramblings that are inaccurate or don't make sense!!
If you have read this entire post, my sincere condolences and your medal is in the post!