I currently have MiniKeePass on my iPhone and although it does the basic job of storing passwords, I don't have the ability to create random passwords from the app itself and sync the changes to Dropbox.
I'm looking for a subscription-free iOS app that had a clean, modern and beautifully designed UI.
I use 1Password which is available for OSX, iOS, Windows and Android and offers both requirements you've listed (random passwords and Dropbox syncing).
Use KeePass. Does the job for me.
KeePass is a free open source password manager, which helps you to manage your passwords in a secure way. You can put all your passwords in one database, which is locked with one master key or a key file. So you only have to remember one single master password or select the key file to unlock the whole database. The databases are encrypted using the best and most secure encryption algorithms currently known (AES and Twofish).
OS X and iOS come with Keychain built-in. The passwords are automatically synced between your Mac and iOS device. You can access the passwords on both.
I've used a number of the password managers mentioned in the comments, even the insanely overpriced 1Password, and, though some may have slightly slicker UIs than Keychain, nothing compares on iOS with Keychain's auto-fill. The closest match on iOS would be iCab browser's password fill and services option, but to get full functionality you'd have to also be using iCab browser on OS X and the desktop version simply isn't up to snuff.
KeePass is a free, open source, and cross platform password manager, which has the benefit that your password file can be used on multiple devices (Linux, Windows, OSX, iOS and Android). I usually use DropBox to sync the password database between my devices.
Unfortunately the official binary on OSX required an X Server, which made it awkward to use compared to it's competitors.
Fortunately there is MacPass, which is a native port of KeePass for OSX. While it's not as well known as 1Password, it is however free and open source, meaning that you can (potentially) check it's source code, if you don't believe it's secure enough, or contains hidden backdoors.
Also since it's a KeePass port, it can open and edit your MiniKeePass files as well.
Description from the Mac App Store:
Locko is a beautiful and easy-to-use password manager and file vault. You can store confidential information such as web logins, credit card data, computer accounts, software licenses, private pictures and documents, and much more. Locko supports iCloud and Dropbox so you can keep your data synchronized across multiple computers while maintaining absolute security.
• Secure AES-256 encryption
• Configurable password generator
• Web Logins, Credit Cards, Bank Accounts, Software Licenses, and so much more…
• File vault
• Photo album
• Secure Notes
• Custom fields (text, password, date, secure text)
• Search in all fields
• Unlimited item groups
• Browser extensions for Safari, Chrome, Firefox, and Opera
• 1 click login for websites
• QuickLook encrypted documents
• iCloud and Dropbox synchronisation
• Import from 1Password
• Export database in JSON format