If you pair bitcoin mining with Distributed Charge - GRID (http://andyschroder.com/DistributedCharge/GRID/Overview/) and a lightning enabled mining pool payout, the miner can collect their revenue and pay for their expenses directly with lightning without needing to make extra on chain transactions. Mining pools can afford to make much more rapid payouts with off chain transactions. Making much faster payments for your expenses with Distributed Charge - GRID allows a miner to operate a bit leaner than they otherwise (they will collect revenue and pay their main expense nearly instantly) would and will allow them to take part in the future where we have a truly decentralized energy grid where instantaneous pricing is based on the price your neighbors are willing to buy and sell energy at right now.

I'd say that will be a big incentive to run a lightning node as a miner, but not necessarily an advantage over players in the marketplace. In a bitcoin only economy, everyone should be collecting their revenue and paying their expenses with lightning, but right now, miners can only collect their revenue in bitcoin, so the will be early adopters