what must be the size of a texture for a specific object?

A very basic, but important question of which I cannot find an answer.

When creating an object, whatever it maybe, how do you determine the size of the texture you want to put on it?

Texture size depends on the detail you need for the object, and possibly the most typical uses for it.

For large floor prims I use 512x512. For "edging strip" faces I might use 32x512, for a door maybe 64x256. For a repeating wall texture, maybe 256x256 will do.

Keeping horizontal and vertical sizes to powers of 2 is good- SL will force this on you anyway so it's better to do the resizing in your own paint package than trust to the SL uploader.

Ok, but what if, for example, I have a cube of 3x3m .... what would be a good texture-size??

The rezzed size is completely irrelevant. What matters is how close people will be to it when they look at it, and how important the detail is to what you are doing.

Read the texture tips that are stickied at the top of the texture forum. It explains it in detail. But in short, the closer you are, the more pixels you will see.

A 3M cube seen from 128 meters might be a single pixel on my screen. No matter how high-rez your texture, that won't change.

That same 3M cube at 1M away fills my screen, and my screen uses more pixels than you can put in an SL texture to render it. The texture looks grainy no matter how good it is.

My rule of thumb is that for most things, 256 x 265 is plenty. For large wall or floor areas that will be a single texture, I may use up to 1024 x 1024. Small detailed things like rimgs don't need more than 512. You can't get close enough to matter.

You want to be a miser with texture sizes. Make them as small as you can get way with. Large textures take longer to rez.

Time to rez is a good point. When we built an island full of houses and stuff, we stuck mostly to a common set of mostly 256x256 tileable textures. You basically never had a gray moment the whole island over almost from TP in.