There are 2 options with their pros and cons:

1. Use signatures
 - Every function in your smart contract must have `signature` parameter. 
 - People who want to interact with the smart contract must sign the function parameters with their account's private key and send it to the smart contract owner (via any communication channel). 
 - The owner then submits the parameters along with the signature to the blockchain, paying for gas. The signature guarantees that the message was approved by the user.

2. Refund used gas at the end of the transaction. A modifier `refundGasCost` can be used for this (see below).

*Below are more details for each option:*

---

**Using signatures**

Here is a simple `ReceiverPays` contract that allows making the receiver of the payment pay for gas:

    pragma solidity ^0.4.20;
    
    contract ReceiverPays {
        address owner = msg.sender;
    
        mapping(uint256 => bool) usedNonces;
    
        // Funds are sent at deployment time.
        function ReceiverPays() public payable { }
    
    
        function claimPayment(uint256 amount, uint256 nonce, bytes sig) public {
            require(!usedNonces[nonce]);
            usedNonces[nonce] = true;
    
            // This recreates the message that was signed on the client.
            bytes32 message = prefixed(keccak256(msg.sender, amount, nonce, this));
    
            require(recoverSigner(message, sig) == owner);
    
            msg.sender.transfer(amount);
        }
    
        // Destroy contract and reclaim leftover funds.
        function kill() public {
            require(msg.sender == owner);
            selfdestruct(msg.sender);
        }
    
    
        // Signature methods
    
        function splitSignature(bytes sig)
            internal
            pure
            returns (uint8, bytes32, bytes32)
        {
            require(sig.length == 65);
    
            bytes32 r;
            bytes32 s;
            uint8 v;
    
            assembly {
                // first 32 bytes, after the length prefix
                r := mload(add(sig, 32))
                // second 32 bytes
                s := mload(add(sig, 64))
                // final byte (first byte of the next 32 bytes)
                v := byte(0, mload(add(sig, 96)))
            }
    
            return (v, r, s);
        }
    
        function recoverSigner(bytes32 message, bytes sig)
            internal
            pure
            returns (address)
        {
            uint8 v;
            bytes32 r;
            bytes32 s;
    
            (v, r, s) = splitSignature(sig);
    
            return ecrecover(message, v, r, s);
        }
    
        // Builds a prefixed hash to mimic the behavior of eth_sign.
        function prefixed(bytes32 hash) internal pure returns (bytes32) {
            return keccak256("\x19Ethereum Signed Message:\n32", hash);
        }
    }

More details can be found in this article https://programtheblockchain.com/posts/2018/02/17/signing-and-verifying-messages-in-ethereum/

The limitation of this approach is that if your smart contract needs to interact with other contracst, then they must also implement the same pattern with adding signatures to every method.

---
**Refunding used gas to the transaction sender**

It's not an ideal solution but you can refund gas cost to the transaction sender. You can do this with a modifier:

    pragma solidity^0.4.11;
    
    contract SomeContract {
        
        event SomeEvent(address sender);
        
        // Need to allow depositing ether to the contract
        function() public payable {
        }
    
    	modifier refundGasCost()
        {
            uint remainingGasStart = msg.gas;

            _;
            
            uint remainingGasEnd = msg.gas;
            uint usedGas = remainingGasStart - remainingGasEnd;
            // Add intrinsic gas and transfer gas. Need to account for gas stipend as well.
            usedGas += 21000 + 9700;
            // Possibly need to check max gasprice and usedGas here to limit possibility for abuse.
            uint gasCost = usedGas * tx.gasprice;
            // Refund gas cost
            tx.origin.transfer(gasCost);
        }
    
    	function doSomething() external refundGasCost {
    	    SomeEvent(msg.sender);	
    	}
    }

Refunding in this way implies some overhead: at least 9700 gas has to be payed extra for the `transfer` function call inside `refundGasCost` modifier. Also gas for other opcodes in `refundGasCost` should be added to `usedGas`.

Also the above code is potentially vulnerable to reentrancy and other attacks. I provided it only as an example and didn't test it thoroughly.


----------
https://github.com/ethereum/wiki/wiki/Design-Rationale

> Requiring transaction senders to pay for gas instead of contracts substantially increases developer usability. Very early versions of Ethereum had contracts pay for gas, but this led to the rather ugly problem that every contract had to implement "guard" code that would make sure that every incoming message compensated the contract with enough ether to pay for the gas that it consumed.
 
... 

> ORIGIN: the primary use of the ORIGIN opcode, which provides the sender of a transaction, is to allow contracts to make refund payments for gas.