My website hosted on Lightsail goes offline at different times every day.

Based on your description, this recurring issue with your Lightsail instance going offline and requiring restarts is likely related to CPU utilization and resource constraints.

Lightsail instances operate on a burstable performance model with sustainable and burstable CPU utilization zones. When your instance consistently consumes more CPU than the baseline provides, it can deplete its burst capacity, leading to performance degradation that makes your website appear offline or unresponsive - even though the AWS console may still show the instance as "Running."

Here are some potential solutions:

1. Upgrade your instance: If you're consistently exceeding your current plan's capacity, consider upgrading to a larger Lightsail plan with more RAM, vCPUs, and higher baseline performance.

2. Optimize your website: Remove unnecessary plugins, optimize your database, compress images, and implement caching solutions to reduce resource consumption.

3. Monitor your metrics: Regularly check your CPU utilization and burst capacity using Lightsail's monitoring tools to identify patterns and potential issues before they cause downtime.

4. Stop and start instead of reboot: Interestingly, stopping and starting your instance (rather than just rebooting) can sometimes resolve persistent issues as this action effectively migrates your instance to a different physical server.

5. Consider offloading content: For WordPress sites, you can use plugins to store media files on S3 or Lightsail buckets to reduce the load on your instance.

If these solutions don't resolve the issue, you might want to consider migrating to a more flexible service like Amazon EC2 that offers more customization options for server resources.
Sources
Recurring Website Access Issues with Amazon Lightsail Instance | AWS re:Post
Website Capacity Issues on Amazon Lightsail | AWS re:Post
AWS lightsail need reboot every ten hours | AWS re:Post
AWS Lightsail Gateway Timeout/Site unreachable | AWS re:Post