A hosted wallet, which will involve a wallet running on a server that has all the wallets is a bad idea in general. People will not feel safe as they have to trust a third party with their coins and you'd have to implement a whole backend system.
Piggying off Coinbase or a Blockchain is a bad idea, the services themselves may send you a Cease and Desist as you may send a lot of requests and end up swamping their servers, also you'd be copying the functionality of their apps.
You can write a full node, but a mobile phone may not have enough ram and storage to efficiently and quickly do that.
The only nice, viable and realistic approach is creating an SPV client like Electrum for the mobile platform. SPV clients aren't full nodes, they put a certain amount of trust on third parties, but the amount of trust is incredibly small and always double checked. Their main method of work is just requesting data from servers, etc. You can use the Electrum protocol instead of rolling your own as well, you won't need to host servers as well doing that as the community runs Electrum nodes themselves.