I'm not too sure how you'd be able to do this - unless there's some sort of log in the distributed system manager (which allows you to view shared variable information that the data dashboard is based on).
Typically you would do some sort of request/response for this - send some data and record the timestamp and then when you get the reply take another timestamp and that gives you your round trip time. Because the data dashboard is a fixed app (i.e. you can't modify/program it) there isn't much you can do on that side.
Perhaps you could do some measure of the latency using a camera? Take a video of you sending some data and the response on the application and then see the difference (i.e. in number of frames) between the two.
If you look at the links in my signature - I was able to get pretty good latency (good enough for Tetris) using WebSockets and a mobile browser - maybe you could do something similar?
The network latency (not taking into account latency for the shared variable engine etc.) is obviously very dependent on how you are connecting your mobile to the application:
WiFi on local network: ~5-10ms
3G: ~100-150ms
4G: ~30-60ms
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