Does the time during the day or night when one has a dream have any influence on how meaningful it is? I've heard it does, but have no good source, and wonder whether it's true.
Yes. The Gemara says (Berachot 55b) in the of Rabbi Yohanan that three dreams are fulfilled and of them is a dream that happened early in the morning. Maharsha says there that the reason is because all the things that happened the day before are usually dreamt about late at night and in the morning the mind is fresh to have a "real dream".