You mentioned that this is a Rails site. Rails allows you to set the SQLite retry timeout in your database.yml config file:
production:
adapter: sqlite3
database: db/mysite_prod.sqlite3
timeout: 10000
The timeout value is specified in miliseconds. Increasing it to 10 or 15 seconds should decrease the number of BusyExceptions you see in your log.
This is just a temporary solution, though. If your site needs true concurrency then you will have to migrate to another db engine.