In Android design guidelines (Settings, Writing Style), it's discouraged to refer to the user:
Don't refer to the user. For example, for a setting allowing the user to turn notifications on or off, label it "Notifications" instead of "Notify me".
Don't use: me, I, my, mine. Use: you, your, yours
I know that Google doesn't follow this guideline on its own login pages. Still, how can you reword "remember me" so it doesn't refer to the user with a first-person pronoun?
A checkbox with the label
Stay logged in
When it comes to Android apps, most of them keep you logged in by default. Remembering login information makes more sense on the web because it might be a shared computer or/and because browser sessions are expected to close all logins.
How about:
Remember Login
In this case you are saying explicitly what the action would do, rather than referring to the user themselves.
Alternatives:
Trust this device
Trust this device for 30 days
Stay signed in
