A signature colour?
One option is to choose a 'signature' colour and get it on all your suits. It helps if this is something a bit unusual like bright purple or lime green, but it could be anything really. Certainly, having a signature colour is a nice way to tie your whole wardrobe together, and it also saves difficult decisions. However in some ways it can also be a waste as picking a lining colour that really complements the fabric is a way to make your suit look that little bit better, and a signature colour is unlikely to do this.
A matching colour
Another relatively easy option is to have a 'matching' colour. Two of my grey suits have grey or silver linings which essentially match the fabric, and my tweed jacket from A Suit That Fits also has a 'matching' brown lining (although this is only because they, quite irritatingly, only give you a 'matching' lining as standard and change £20 or more for anything else.)
A matching colour is a safe enough bet, but it's a little bit boring. I always think that having a slightly unexpected colour for the lining, to be glimpsed when the jacket swings open, is much nicer than simply matching the lining to the fabric.
A complementary colour
This is probably the best choice, although it really covers a multitude of options. It's also where I struggle most, as picking a lining that goes with the fabric, but doesn't actually match it is a little tricky. One example, I suppose, is my dark red lining with my black dinner jacket, but that may be cheating as black famously goes with almost anything.
Perhaps a better example is the light purple lining in my dark blue self-stripe suit from ASTF. I can't take much credit for this one either as ASTF has a feature that suggests lining colours to go with the fabric you have selected, and I simply took their suggestion. It's a good one, though, and it does what lining can do well: being bolder and more unusual than the colour of the fabric itself, but still clearly related to the suit as a whole.
A Contrasting Colour
Anothing lining colour that I very much like is white in a dark grey suit. I have a double-breasted charcoal grey pin-stripe with a stark white lining and I think it looks fantastic. A lining like this would look great in any dark suit, perhaps even a black dinner suit. Equally, a black or dark grey lining might look really nice in a cream suit or dinner jacket.
Some people probably find choosing lining colours easy. Unfortunately I'm not amongst them, but it's definitely something worth giving thought to, as people will see the lining more often than you might think; as the jacket opens, around your sleeves, inside the pocket flaps, and so on.