Having ruled out the power sources and storages based on chemistry, providing power appears to be the major issue for the century-lasting electronics. But let's talk a bit also about those "less important" issues.
- ELECTRONIC PARTS - while this is seldom stressed, the electronics parts industry appears to have a pretty impressive record as far as long-term reliability is concerned; and it also has improved much during the decades it exists, mainly due to pressure from automotive and aerospace industries. While restorators of early electronics (mainly broadcast radio receivers) have to cope with many various failed parts - resistors, capacitors, electromechanical parts (and of course the inherently failure prone electron tubes/valves), in "solid state" electronics usually the electrolythics are the central point of failure (and we've already identified that as a problematic type of part). With careful fully automated manufacturing from pure materials in clean room environment, parts now have typical guaranteed life of 10-20 years at 85-105°C, so it's very likely they will be OK after a century of operation at room temperature.
- PCB, SOLDERING - the glass-epoxy combination appears to last for ever, judging from vintage electronics. It would be better to avoid lead-free solder, even if vendors swear the whiskers and tin plague have been wrestled down, it's better to stay conservative. Anyway, who would dump a gadget to landfill, if it's perfectly OK even after a century and can be continuously used or resold for good price? ;-)
- MECHANICAL MATERIALS (enclosure, PCB support) - while for most gadgets' encapsulation, plastic is the material of choice; I'd rule that out for this particular purpose. There is a very wide variety of polymers, copolymers and similar materials, often enhanced by various additives and fillers. Unfortunately, we've seen too many of them to get discolored, brittle or soften, and break apart after several years. Their internal structure changes upon UV irradiation, exposure to heat or chemicals/gases (sometimes those commonly used in households e.g. for cleaning), or sometimes even just like that, spontaneously. While there are plastic materials which are deeply investigated also for long term durability, there's just too many of them to say safely this is one which would last. So, in this particular case, it's wise to be conservative and use materials which have been proven by centuries of use: metal, ceramics, glass, and - with some care - wood.
- SIZE, MECHANICAL CONSTRUCTION ISSUES - as these gadgets are going to be part of a house, their size does not need to be very constrained. Some of them may be built into walls or "integrated" with the house in some other way, so a couple of extra inches in every dimension does not really count. What does count, though, is their appearance and design, this is something which has to be considered carefully; it's not going to be your average white or black box. There may be one great opportunity in having wireless and maintenance-free electronics, namely that it could be hermetically enclosed e.g. in a glass or combined glass/metal enclosure (with careful choice of sealing method and mainly material - rubbers are out). This would also increase resilience to any adverse outer environmental impact (humidity, gases and chemicals, rapid temperature changes).
- COST - as these are to last and serve for an order of magnitude more time than the common stuff, the budget may be also roughly an order of magnitude higher.