What words did the LED clock display? Was it a string of letters, such as H, L, O, S, and the like? Do they randomly appear and are transient [only there for a few seconds], or do they stay resident for a certain time. Does unplugging the clock and repowering it bring up numbers or letters.
Sounds like the place might possibly have a carrier current type system that is interfering with these clocks' display ICs [Integrated Circuits - AKA "Chips"].
Carrier current systems are the controllers that use the AC power branch circuits for the control signals, as well as the power [a different kind of wireless control]. Plug-in Intercoms work this way, along with some lighting control systems sold retail for home use and homeowner installations.
Remember the clocks in School when the time got near the hour, and they would run fast until the minute and second hands were on the 12? These use Carrier Current type of controls.
Another source of trouble could be RFI on the power system, but I have never seen it to that extent.
RFI could be from a variety of sources. Local Radio broadcasts would be the first and most common. These would be either from a commercial AM or FM broadcast output antenna being close to the location [this can be verified visually or by listening for interference on the telephone]. Another Radio source would be a HAM Radio operator, using the band that has the best carrier possibilities for the Radio users, but the worst effects to TV, Telephone and also the local AC power [I do not recall the band, but I think it was 30 MHz "10 meter band"]. My old neighbor was a HAM user and I had to filter the heck out of things to keep the "gremlins" out!!
Other interference sources could be a PC with a power supply that is ready to fry, Inductive components with a ridiculously bad quality issue [harmonics and such], even an alarm system that is wireless.
If the local power company does not find any quality issues on their end, see if you can locate anything suspectable. If you have an oscilloscope, see what the AC wave looks like at different locations and times in the house. Run the problematic devices with the scope connected also to see how it's looking. I would look for transient waves past the 9th harmonic, plus the way the fundamental is distorted [see if it is too sharp at the peaks and stuff!!]. Any abnormal harmonic could be suspect if it gets into the timer crystal [or 555 timer IC].
If you have access to a graphing voltage meter, run it for a day or so to see what the transients look like [transient voltages and currents].
I have posed this problem to a few of my Engineering buddies, but we all have the same thoughts. I'll need to get in contact with a few Electronics Tech friends of mine to see what they think, however they are away for the holidays.
I have got to find out what is happening here! If I ever run across this again, it might be helpful
The end result is most likely going to be filters, unless the utility company finds un-normally low voltage or brownout problems in their equipment and decide to correct it.
Find out what else [if any] is malfunctioning at this house, such as Incandescent lamps being very dim, or very bright, along with them "singing" [dimmers or AM Radio interferences]. Check for anything that is abnormal, plus check the things that the neighbors are doing, such as home work shops, arc welders, Radio, etc. and see if this might contribute to the problems.
Sorry to give you so much information, but I really have no clue what to do other than a step - by - step approach to see what is going on [kind of stumble across the culprit]. I would like to know what the end result is! and if I was anywhere close.
Keep us informed.