I'm not sure it would.
The question is where to place it for maximum benefit, but the line doesn't pass that close to potential customers, and skirts around the eastern edge of the town.
Others might have a higher threshold for walking distance to a station, but as the line leaves the port it passes the Trumacar estate (low income), a caravan park, and a modern housing development that isn't even served by bus. I could be wrong, but I don't imagine many users, its just not central enough. There is a bus, the 2x, that links Heysham direct to Lancaster without passing through any Lancaster or Morecambe suburbs.
How many would walk ten-fifteen-twenty minutes, for a fifteen minute train ride, with a longer walk at the other end, when the bus is practically door to door, and twenty minutes max? I'd like to see greater rail use, but I don't see much growth on the Heysham branch.
More services south from Morecambe though, that would drive some growth. I was a regular commuter from Morecambe to Leyland, and that was two changes, and I wasn't the only one travelling beyond Preston from Morecambe everyday. The sparks effect of wires to Morecambe, and direct commuter services to Manchester would soon transform Morecambe's fortunes.