As I've probably mentioned (many many times) before, I have really dry and sensitive skin, especially around my eyes (which is prone to eczema too) so having a very gentle- but effective- make up remover is pretty important to me.
After my previous remover ran out, I took a small bottle and filled it with Extra Virgin Olive Oil (I've heard this is the best kind of olive oil for the skin) straight from the bottle in the kitchen :) You can get it very cheaply from the supermarket (about £3 for a litre!) and for eye make up removal purposes you need very little, so even a 100ml bottle should last a really long time.
I really really recommend trying olive oil as a make up remover, it is SO gentle and effective- it's especially good for those with sensitive or dry skin, but I think absolutely everyone should try it!
p.s.. Extra Virgin Olive Oil also makes a lovely salad dressing :p