Following Simon Wilsons suggestion on the current Marble Sink thread a discussion about techniques using the much maligned ladders might be useful entertaining hopefully without any slagging off.

I started caving before srt was imported and ladders were all we had. I have used them with and without lifelines on short and medium pitches.
Lifelines were always full weight (No 4 nylon laid and then 11/10mm kernmantle) Would be interested in views on using much lighter ropes, perhaps with a simple friction device, to avoid excess bulk and weight of the traditional lifelines.

Jopo

If you're belaying properly it should only be a top rope slump rather than a lead fall, so I'd have thought 9mm should be fine

If you're looking to go down the weight saving route, considerably skinnier ropes are around that are rated for this type of application.
Any climbing (i.e. dynamic) half rope should be fine, you can choose from the terrifyingly thin such as Mammut Twilight (7.5mm, 38g/m) or Beal Gully (7.3mm, 36g/m) to a wide selection of more substantial and abrasion resistant options. Personally I'd probably choose a unicore rope in the region of 8mm for lifelining.
Well, except that I dislike ladders and go for SRT with only very rare exceptions.

EDIT:
Just to quickly point out the obvious, whilst those seriously thin ropes pack down really nicely and weigh nothing, they also don't work very well with the type of equipment we usually use in caves. A no-return pulley would work, but might prove a bit unfortunate when the person below gets stuck under a waterfall and the gurgling sounds start. The likely solution is a dedicated small diameter belay device or maybe a monster munter/ super munter.