Other than the dash lighting up, there are no indications of trouble - no messages in the center display for example? On the START switch, there is a small LED light... what color is it when the car fails to start? RED/AMBER or BLUE? If Blue, the car thinks you want the dash ON but don't what to start the engine... With engine running, the LED should be off.
If you push the key fob into the slot below the stereo (earlier model year cars) or hold it against the START button (later model years witihout the slot) while pushing the button, does that make a difference? In theory the dash shouldn't even light up if the car can't detect the fob; putting the fob into the slot or holding it against the button is supposed to help if the car can't detect the fob at all. Still, even if your dash lights up (indicating the fob can be seen by the car) it never hurts to try the backup method. If the backup method works, your car needs dealer attention regarding the fobs or the multitude of electronic bits that get into the alarm and start-up system.
How far does the brake pedal move when you are trying to start? If the power brake assist isn't working, the pedal will be very stiff and won't move down enough unless you really stand on it. Once the engine starts, if the brake pedal then mushes down a lot, that is a sure indicator that the brake vacuum booster unit, or the hose feeding it, or check valves in the input, are bad and need to be replaced ASAP. It's a safety thing - the brake power assist will be lower than normal and might not exist in the event the engine stalls. Basically, engine vacuum is supposed to be "trapped" in the booster assembly; that vacuum should remain trapped for at least a day (should really be a couple of days) without leaking away. A brake pedal that is very stiff as soon as you press it indicates loss of booster vacuum from leaks. Note: to enable the start sequence, the brake pedal has to be depressed a tad beyond what it takes to trigger the rear stop lights. If you can't see the brake lights reflecting off the garage wall or a car parked behind yours, either you aren't pushing the pedal far enough of the switch is still bad.
You mentioned you tried putting the shift lever in Neutral as well - so you seem to be aware of the "Park/Neutral start switch" failure that can affect many auto transmission cars... still, while trying to start, wiggle the lever and/or push it forwards or aft in the Park and Neutral positions - don't just put the lever in eitehr position and leave it/let go of it. Basically, try anything you can think of to deal with a bad/flaky switch. If that makes a difference, the shift lever assembly By the way, Hyundai manuals call this the "Transaxle range switch" in the service manuals.
And double-check the battery cables plus the ground in the trunk. Even though the dash lights up, issues in the battery wiring have caused all sorts of goofball issues for other folks on this site. It's easy enough to check. Crud under the ground bolt in the trunk is common; another issue is the positive cable clamp may not properly grip many replacement/
aftermarket batteries... adding a shim of some sort cures those issues.
If you have some mechanical skills, locate the starter motor itself on the engine. It'll have a beefy wire feeding it; that's the +12volt feed for the motor. There will also be a smaller wire going to the solenoid assembly - the smaller cylinder part of the starter motor assembly. Put a voltmeter on that wire (and to battery negative post or any decent ground point) and look for +12volts when a helper hits the START button. If no voltage is present, the bug is in the car electronics; if voltage is present then it appears the starter solenoid is dead. You can also use a small 12 volt light bulb in this wire - instead of a voltmeter - to make the test... a bulb like a turn signal bulb works well. Don't try a large bulb like an old headlight bulb though.
The engine-start sequence is controlled by the "Power Distribution Module" (PDM) under the dash, mounted vertically next to the stereo stuff, above the metal reinforcing plate; it has two wire connectors feeding it. It drives a handful of relays in one of the under-hood boxes for the equivalent of mechanical key car's ACC, ON, and START positions; they're labeled STARTER, ACC, IGN1, and IGN2 or ESCL-something. It also triggers the steering
wheel unlock - do you hear that whiz-whiz sound when the car won't start? The PDM is actually two computers inside; if either fails the car can be shut off by holding the START switch for a few seconds, or pushing it three times, but it won't allow the engine to re-start. Besides the relays, a 15 amp and a 25 amp fuse (the manuals don't clearly specify where each fuse is located...) feed the PDM so verify those are good and have tight connections. The 25 amp fuse might be labeled ESCL 1 in one of the dash panel fuse boxes; the 15 amp should be in one of the underhood boxes I believe. There is a 10 amp fuse in one of the dashboard panels labeled START; this actually feeds the transaxle range switch. Unplug each one and verify clean contacts. Also, the BLOWER relay (interior a/c fan motor) might be same type/size of relay and, if so, can be swapped with others as a test. The rear window defroster ("heated glass") relay is in that box too; if it is the same physical size then it too is a swap-test candidate.
One other thing is if the "imobilizer" circuit has been activated for some reason... this prevents the starter motor from engaging. Roll a window down, turn the car OFF again, and hold the fob out the window. LOCK and then UNLOCK the car. Bring the fob inside and try starting now; if the engine starts reliably this way then something is tripping the imobilizer.
mike c.
