Given their widespread uses on phones, the overall average is likely very small. KBs even.
A more relevant question might be: what would the average size of SQLite databases for web type (or even this specific use case) applications. I don't know, but 10s or 100s of MBs might not be a bad guess.
However, would those 10s or 100s of MBs be something web sites would just like to share in their entirety to the client to query as they wish? At least many commercial services would prefer to keep most of the data secret and even gain insight from the access requests.
But for a more open or a hobbyist project where minimizing amounts of data transfer is less important than minimizing the amount of doing more work (server-side code), then this seems like a decent solution.
It is also worth reminding that this solution only practically works for read-only databases.
A more relevant question might be: what would the average size of SQLite databases for web type (or even this specific use case) applications. I don't know, but 10s or 100s of MBs might not be a bad guess.