I've never heard of HDTVs not picking up signals. That's really odd. Are you sure it's the connections? Like, does the screen blip at least when you turn your N64 on? If there's some sort of movement, then there's probably a connection being made from the N64 to the TV, but not from the cartridge to the N64. Which probably means you need to clean out the connectors on your cartridges and your N64. I had problems with my N64 suddenly not working anymore, then I noticed all the dust that had accumulated over years. I too keep all my stuff in tip-top shape, but I'd still leave my cartridges laying about occasionally, and I used to haul my N64 around with me. Since cleaning everything, it all works wonderfully, and I store all my games in a place where dust can't get at them.
I can't recommend playing old consoles on an HDTV, though. The oldest I go is 480p GameCube games, and occasionally 480p PS2 games, though there are few of those that look stellar (Jak II for instance looks great, but Ratchet & Clank: Going Commando looks real blurry still). All my interlaced GameCube games I play on an SDTV via S-Video, and that looks way better than playing them on my 52" Toshiba REGZA—and that's even with adjusting all the video settings to get the best look.
Though I would be interested in one day buying a few official Nintendo SCART cables to plug my SNES, N64, and GameCube into my HDTV. Of course, I'd need to buy a SCART-to-component adapter, and that should carry the signal over just fine with no degradation. I'm guessing old consoles using SCART would look rad on an HDTV. And if they don't, I intend on one day getting an EDTV CRT (acronyms!) that's widescreen and has a component input, so I could enjoy some of my old consoles in super awesome quality, since S-Video, while loads better than composite, still has some weird issues I'm not happy about, most notably dithering (at least I think it would be called dithering).
Ideally, all my favorite N64 games would be released on Nintendo's Virtual Console, since they're all 480p (and I play my Wii on my HDTV via component); most N64 games have half that resolution (sometimes a bit less than half), and are of course interlaced. And the games that use the 4MB Expansion Pak for 640x480 resolution, they often have some very nasty slowdown issues, forcing me to play with low resolution anyway since I'll take smooth, playable gaming over poor framerate any day.