I want a wallet on my Android phone that is rooted. I'm new to Bitcoin and I'd like to know which one is technically secure.
It's a difficult question to answer because an Android rooted device is never going to be secure 100%.
That being said, the safest options will be 2FA enabled wallets at a bare minimum, and multisig solutions being the best bet.
GreenAddress.it allows for wallet functionality on par withBlockchain.info but with increased security through P2Script using ntimelock. It also allows for 2FA: be sure to use TOTP (eg Google Authenticator); do not use the same device for mobile SMS authentication since intercepting sms on a rooted device is extremely simple.
GreenBits is a new app which integrates with GreenAddress and utilises 2FA and P2Script also.
Options like BCI and Mycelium are not ideal for a rooted phone as the security concerns make either choice quite vulnerable to being emptied by malicious software stealing the pin code.