Is it possible to define a texture that is always bright (like some kind of "Full Bright" checkbox)? So the texture is not affected by light sources and always good visible.
No. You just need to use an appropriate shader instead of using one that isn't made for it. Using some shader that takes lighting into account is nothing but a performance hit otherwise.
http://www.unifycommunity.com/wiki/index.php?title=Texture_Only