The message
Quote: MySQL server has gone away
Was returned by MySQL itself. Presumably this is the key to what went wrong, but I have no idea what could cause that. Perhaps your ISP can help here.
Basically, Zenphoto has made a successful connection to the database and issued the query
Quote: INSERT INTO zp_options (name, value, ownerid, theme) VALUES ('Allow_search','1',0,'zenpage-default')
which resulted in this failure.
The above query is presumably from the theme setting its default options. If this is a new installation, that would make senes, but if it is an existing one, the option should long ago have been set. That would be an indication of some other problem with database access.