I am using my mother's Elgin 3400E travel alarm

Because you have an alarm set for 11:58?

https://www.google.com/search?q=elgin+alarm+clock+4037+cancel+alarm

..

you have it set on 24 hour time instead of 12 hour time
it will change over at midnight to military time as that is when the 24 hour period ( military time ) starts

Did you ever figure it out? I have the same problem. I see no one answered you here.

with the switch in the 'lock' position, hit the C/F button (also labeled as the 'min' & 'date' button)

Travel alarm clocks can have varying instructions, depending on the type of clock you are using. Here are some tips for use that will apply to any alarm clock.

1. Power source
   • Travel alarm clocks will need to have some sort of power source. Depending on the type of alarm clock, it might be battery-powered or it might have a cord that needs to be plugged in. Battery-powered alarm clocks can take mini watch batteries, AAA batteries or sometimes AA batteries. Travel alarm clocks, because of their tiny size, usually have the small batteries that need to be bought from a watch store or a dollar store. Before you leave, make sure you have a working power source and can turn it on.
     
   Changing the time
   • After you reach your final destination, power the alarm clock on. Now you'll need to set the time and date correctly. Check a watch, cell phone or television to get the exact time and locate the buttons or dials on your alarm clock. On some, there will be a "time" or an "hour" button you need to press and hold to activate the alarm time change. While you are holding it down, turn the dial, press the hour button or press the (+) button until you notice a time change. On others, simply pressing the hour or minute button will let you change the time shown.
     
   Setting AM/PM
   • The AM/PM option is critical in setting your alarm clock correctly. Make sure the LCD display on your clock has a light beside the correct AM or PM setting, or look for the letters AM or PM on the display. If you get this wrong, the alarm won't go off in the morning!
     
   Setting alarm time
   • To set the alarm time, look for a button or a switch that says alarm. If there is a switch, move it to the alarm setting. If it is a button, you'll need to hold it down until the time starts flashing, signaling time to change the hours and minutes to the desired alarm time, or hold it down while you press the hours and minutes buttons to reach the alarm time you want.
     
   Turn alarm on
   • Now double-check that your regular and alarm times are correct by switching the switch back and forth again, or by pressing the alarm/time buttons long enough to read the time set. There should be a picture of a bell somewhere on your LCD display showing that the alarm is set to "on." If there isn't, look for the on/off alarm button and move it to the correct position.
     
   Music options
   • The last step is finding the alarm wake-up option. Not all travel alarm clocks have this option, but if yours does there will be a switch or button that lets you choose to wake up by a buzzer, the radio, a CD, or another device.
     

Read more: Travel Alarm Clock Instructions ' eHow.com http://www.ehow.com/way_5250617_travel-alarm-clock-instructions.html#ixzz1YfneKhGw

Often the trick is to hold one of the keys (time/mode?)
for about 3 seconds to go to the settings mode.

Usually the problem is a bad batch of batteries. Try taking one from something that is working and see if that is the case.

I have an old Lorus travel alarm LHL008N-F and need to know what battery size it uses.

I need the exact same thing; I just bought one of these secondhand and how does this thing work anyway?

I have a 20 yr old Elgin Anniversary Clock with the Magic Eye battery operated. It chimes 15 mins after the hour. How do I correct this?