Transferring ye olde camcorder tapes to MODERN DVD. Best method?

It's a very painful process without specific hardware. I would just pay someone who has such equipment to do it.

of course it will be possible, but in my experience the audio and video will have offset, depending on how long the videos will be.
If the capture card has an hardware encoder (x264), it will be really worth it

the image quality should be the same

Is your camera like this?
https://video-naar-dvd.be/data/articles/images/lightbox/big/sony-hi8-analoge-videocamera_1038_5.jpg

or your cable like this:
http://www.trait-tech.com/uploads/details/T-CameraC-1005-1__Cable-for-SONY-HI8.jpg

Which exact Sony camera model is it?

have a look at the bottom of the cam, there should be a label with the exact model name. like "CCD TRV85" or similar

OK, your camera has female RCA connectors for its outputs - Black for Audio, and Yellow for Video. The Audio, by the way, is single-channel (NOT stereo), and that's why there is only one.

The inputs to most video capture cards are also female RCA connectors, usually using Yellow for Video (more precisely, Composite Video, which is what you're doing here), Red for the Right channel of stereo Audio, and White for the Left channel. Very commonly, of you have only monaural sound (the Black source), you'd connect it to the Left input channel, but check your capture card's manual. Even if your capture card has a smaller 3.5mm jack hole for its inputs, it very likely comes with a cord or adapter to allow you to arrange inputs via RCA connectors.

So, what you need is a simple pair of RCA cables with male connectors on each end. You could use a common cable with three wires in it - Yellow (Video), Red and White (two stereo channels) and just NOT use on of those colours. You could even use a common two-wire cable intended for simple stereo audio, with Red and White, and just use your own colour coding to make the connection. For example, Camera Yellow (Composite Video) gets cable Red, the other end of Cable Red goes to Capture card's Yellow (Video) input, and use the cable's White for Audio.

There is nothing special inside the cables with the different colours in such multi-wire cables. The colour codings are ONLY there to help you keep your connections straight. I have seen a few sets, though, where the Yellow Video part of the cable set is heavier for slightly better quality on that more complex signal.

By the way, the Sony (and other makers) "Hi 8" system was just a slightly better-quality recording system than the original 8 mm video tape system. It made almost no difference to how the signals were exported from the camera to the TV or recording device; the output still was Composite Video plus monaural or stereo audio. A few cameras also had an S-Video output in addition to the Composite Video (S-Video connectors are different), but most of them also had stereo audio so I suspect your camera does not have this port.

If you really want to have your monaural audio signal from the camera recorded on both stereo channels so that both speakers play the same sound, you can do this. Get a small adapter that converts one audio signal into two parallel outputs that you can feed into both stereo inputs of your capture system.

Glad to hear you are having success.