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".
It also implies that it's not down to you, but to some sort of chance.
Outcomes of our actions are influenced by our actions, and are also influenced by external forces - often ones we don't know about in advance. Believing that fact can plausibly have a negative impact on one's motivation and performance. But there is also a positive aspect to having your locus of control moved outward, other than the whole accurate beliefs thing: a poor outcome is less indicative of poor performance, and a very good outcome doesn't have to reflect an outlier performanc...
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.