If you're looking for a meeting place or a good stop around ½ or halfway between Layton and Elko, you can try West Wendover, NV, which is about 2 hours west of Layton, or almost 2 hours east of Elko.
Other possible places to stop near the midpoint include Salduro, Silsbee, and Fondo.
About ¼ or a fourth of the way, you can stop at Flux, UT. This is almost an hour southwest of Layton.
About ⅓ or a third of the way, you can stop at Low, UT. This is about 1 hour west of Layton.
About ⅔ or two thirds of the way, you can stop at Pilot, NV. This is a little over 2 hours west of Layton, or about 1½ hours east of Elko.
About ¾ or three fourths of the way, you can stop at Holborn, NV. This is almost 3 hours west of Layton, or almost an hour northeast of Elko.
If you drive straight through without stopping at all, the total distance is 249 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 3 hours, 37 minutes.