I don't think there's anything we can do directly with weinre to help out here. But the request isn't precise enough for me to be more specific.
If you are wanting to log data produce on the phone/client, then you can either save it in localStorage or some persistance capability on the phone/client (eg WebSQL, IndexedDB). And then have the phone/client upload it every day or something. Or just send the log messages to a logging service you build yourself or buy (eg, https://www.loggly.com/
). Warning that users of your app may be grumpy if they see you uploading data at times when they're not expecting you to be uploading data.
If you have access to the phone/client that is doing the logging, and saving the log data in WebSQL or IndexedDB, you may be able to use weinre or any other debugger (eg, mobile Safari or mobile chrome debugger) to view the log data "live".