Why not "good luck"?
I think that it feels empty. If you have or have not got luck, then I doubt it will change by me saying so. It also implies that it's not down to you, but to some sort of chance.
Similar with 'I hope things get better'. I'm certainly not helping you.
Instead of 'good luck', I tend to say something like "I know you're a capable person, and I expect you to do well. Do not worry, because I have taken the outside view for you, and told you it looks good."
Instead of 'I hope things get better' I tend to say something along the lines of "I want you to know that I and others are here to help you if you need it, so try not to worry".
Is there any rationalist equivalent of "good luck"? I've tried a few variants, such as "work well", "knock them dead", "we're with you" and certain situation-specific phrasings, but haven't found anything that worked generally - though a hearty "may all the gods of Olympus be with you!" can serve. Not a vitally important point, but it would be nice to have something similarly supportive and yet accurate to say.