Hi Morag. This is a difficult one and I think you need to try listing at different times and see what sort of views you get. The 'suck it and me' methodology!!
There is also a feature on the Craft Cult website that may help (not sure if you have used it) - under 'Heart Charts'. The final chart enables you to see what time of the day your shop is most active. It is using one of the US time zones, though I'm not sure which one.
You could also try Google Analytics - though I haven't signed up to it yet so I'm not sure how much info you get as to the busiest time of day for views etc. I tend to rely on my Activity Feed for indications of when my shop is most active (i.e. hearts).
Personally, I don't worry too much about when I list/renew - though I do try to spread my listings/renewals out so they are not bunched together.
However, art is more of a saturated category so it would be better to try to list or renew when you are most likely to get views. Research on e-bay has suggested that evenings are the best - and busiest of all is Sunday evening from 7-10 pm.
I'm not sure about etsy - some people say that weekday evenings are best - also office lunchtimes (approx. 1 pm).
Most of my buyers are in the US so my guess would be that it's probably best to try to synchronise with those times in the US (especially the east coast). But I also find that my shop gets a lot of views on a Sunday (possibly because I sell crafting stuff - and people are feeling more crafty on a Sunday).
There is also a site (not sure which one - but you could ask on the fora) that enables you to pre-schedule your etsy renewals - so you can renew during the night when you're in bed, if that suits you better. However, these 'hack' sites often have 'buggy' elements to them so I'm not over keen myself.
I usually try to renew something before I go to bed - in the hope that someone in the US sees it during what is their evening (as most of my buyers are in the US). But I have also listed things mid-day before now and they have sold immediately to buyers in Australia or Japan (i.e. their evening). So it's anyone's guess really! Just try to spread them out a bit would be my advice.