If you're looking for a meeting place or a good stop around ½ or halfway between Elyria and Cleveland, you can try Westlake, OH, which is about 18 minutes northeast of Elyria, or about half an hour west of Cleveland.
About ¼ or a fourth of the way, you can stop at Avon, OH. This is about 10 minutes northeast of Elyria.
About ⅔ or two thirds of the way, you can stop at Lakewood, OH. This is about half an hour northeast of Elyria, or about 13 minutes west of Cleveland.
If you drive straight through without stopping at all, the total distance is 30 miles. The distances shown above don't include local roads to get into each town, so if you're planning a long road trip, make sure you factor in traffic and road conditions, rest stop breaks, and interesting detours along the way. If you're in a rush, you can do the trip with zero stops, and your total travel time would be around 38 minutes.