Airbags. Electronically controlled.
Electric power steering. Need I elaborate on the "electronic" aspect?
Continuously variable transmissions. Automatic transmissions with more than 3 speeds. CVT was patented in 1886, first used on a car in 1896. 4 speed Hydramatic was an option on 1940 Oldsmobiles, used in some WW2 military vehicles. What automatic transmission with more than 4 speeds doesn't depend on electronic controls?
Front wheel drive with transverse engine mounting. This technology was basically confined to the BMC Mini 50 years ago, but is now nearly universal among passenger cars and crossover SUVs. DKW 1931, Saab 1949, Mini 1959. All much older than 50 years. Admittedly much more popular today than 50 years ago.
Radial tires. Patented in 1915, Developed and commercialized by Michelin, original equipment on 1948 Citroen. Again much older than 50 years.
Suspension in general. With the exception of a few outliers (Corvairs, VWs, Porsches, Corvettes, Jaguar E-Types), solid rear axles and oxcart leaf springs were more or less universal 50 years ago, even on pricey European sports cars; now even many full-size SUVs have sophisticated multi-link rear suspension (and those full-size SUVs honestly handle better than those European sports cars did). Did I mention that modern suspension doesn't need to be lubed every 10k miles and rebuilt every 60k miles? If it was in production use over 50 years ago (Corvairs, VWs, Porsches, Corvettes, Jaguar E-Types, etc) then it's not a significant mechanical advance made during the last 50 years.
Variable valve timing and lift; more specifically, systems such as BMW Valvetronic which eliminate the need for a throttle plate. Valvetronic = Valve + electronic. "This highly advanced technology replaces the conventional throttle butterfly with a electrical mechanism that controls the amount of lift of the individual intake valves on each cylinder." - BMW blurb. Did someone mention "electronics" earlier?
Turbochargers that really work under normal driving conditions. Patented in 1905, very common on aircraft and diesel engines since WW1. First used on production automobiles (Oldsmobile Jetfire) in 1962. Admittedly more common than previously (like FWD).
Traction and stability control. (OK, it's mostly electronic and works in conjunction with the ABS, but I think it still counts.) Interestingly, stability control statistically delivers a real-world safety benefit that ABS promised but didn't deliver. Electronics again.
All-wheel-drive systems that really work under normal driving conditions. A standard 4WD with a center diff. Miller AWD competed in the 1931 Indy 500, standard on Jensen FF in 1966, commonly used on military trucks since the early 1900's. Mechanical technology that's been in use for well over 100 years.
Multi-piston swash-plate A/C compressors. Even a lot of auto enthusiasts are probably saying "Huh?", but have you ever driven a 1960s or 1970s car and noticed that when you turn on the A/C, there's strong vibration and the engine loses so much power that you'd think the handbrake has just been pulled? Ever wonder why new cars aren't like this? Now you know. Sorry, I rate advances in air conditioners as right up there with advances in power mirrors, radios, power seats, power windows, and curb feelers as "nice to have" but not an advance in a primary mechanical system that would inspire me to buy a new car.
Lastly, although advances in fuel injection and ignition have arguably been incremental, the sum total is pretty amazing; consider that you can buy a fairly inexpensive sedan with a naturally-aspirated 2.0L engine that delivers 155 hp, 150 ft-lb torque, and 41 highway MPG by running 13.0:1 compression on 87 octane gas (Mazda3). And it runs well in all weather, at any altitude, without having to fiddle with the carburetor jets and ignition timing. And without tuneups every 15k miles. And the computer tells you what's wrong with the system when it breaks. Pretty sure I mentioned electronic fuel injection and ignition earlier.