It's close to the latter, but a little wider in meaning. Here are some dictionary definitions:
- Everyone gets a chance eventually: Don't worry, you'll get chosen for the team. Every dog has its day. You may become famous someday. Every dog has his day.
- Everyone is successful during some period in their life: He'll get that promotion eventually. Every dog has its day.
- Even the lowliest will sometimes come to the fore, as in: They may not listen to me now, but just wait, every dog has its day.
This proverbial saying alludes to the lowly status dogs once held. [Mid-1500s]