I installed the GE Basic Timer 15070 ok. I set on and off times followint the insdtructions several times. The time will tunr the light on at the set on time but it never turns the light off at the set off time.

Copy following link for timer wiring.
Your box does not have neutral wire
The GE 15070 7-day timer wires same as GE 15312 7-day sunsmart timer
http://waterheatertimer.org/How-to-wire-GE-15312-timer.html

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I don't have repair for defective timer:
If Load and Line wires are reversed, that could explain problem. Reverse timer red and timer black wires. Symptom of this problem: the clock dial stops rotating after tripper pin.

Troubleshoot run-through:
1) Use fingers and manually rotate timer dial several times and watch if Load turns ON-and-OFF
This will tell you if load is operational, and pins are set correctly.
2) Set slide switch to Timer.
3) Make sure timer is wired correctly. This timer needs neutral wire to operate clock motor. Open following link to see illustration and read about wiring. Line and load wires must be correct.
http://waterheatertimer.org/images/GE-15070-wiring-1-500.jpg
http://waterheatertimer.org/Woods-timers-and-manuals.html#59017
4) Check that wire nuts are attached very tight and there are no loose wires.

Additional resource:
http://waterheatertimer.org/How-to-set-timers.html

The white wire has to be connected to neutral wire for timer clock to operate.
Open following links to help with wiring:
http://waterheatertimer.org/GE-timers-and-manuals.html#15070
http://waterheatertimer.org/images/GE-15070-wiring-1-500.jpg

If timer dial is rotating and keeping good time, then clock motor is good, and wiring is correct.
If timer turns the load on-off, then set of 156EB945A trippers are good.
http://waterheatertimer.org/pdf/Intermatic-e1020-english.pdf

I believe the manual control button reverses the program.
So if timer is ON, and manual is pushed, then timer turns OFF, and timer resumes programming at next tripper.
So the manual override is not an on-off switch.

All plug-in timers work same way, and most of the hardwire timers and in-wall timers.
The exception is the analog GE 15070, Woods 59017, Tork 701A, Intermatic E1120, Intermatic KM2 series. These timers have a true on-and-off switch that slides back and forth.
http://waterheatertimer.org/Programmable.html#15070
http://waterheatertimer.org/images/GE-15070-mechanical-timer.jpg
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You don't say if this is new timer installation.
Or if timer was working before and suddenly stopped working.

I also own Woods 59017 timer from Home Depot.

1) New timer installation:
Open following link to access manual and read wiring instructions.
http://waterheatertimer.org/Woods-timers-and-manuals.html#59017

2) Old timer that stopped working:
If timer dial does not rotate, then the timer is broken, does not have power, or is wired wrong.
If timer stalls when it gets to a pin, then the clock motor has gotten weak, and is overheated and will stop working.

3) Buy new GE 15070 timer from Home Depot for $18
http://waterheatertimer.org/GE-timers-and-manuals.html#15070
4) Buy new Woods 59017 at Amazon for $11 plus shipping
http://www.amazon.com/Woods-59017-Mechanical-Repeats-Switch/dp/B002MO3LWQ

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You posted under Grasslin in-wall mechanical timer.

Instructions for this timer are nearly identical to GE
http://waterheatertimer.org/GE-timers-and-manuals.html#15070

First, the timer has to be wired correctly, so review wiring instructions shown on link above.
Next, the pins have to be set, so read instructions and look at image of pins shown on link above.
Last, the timer switch has to me set. I believe timer switch is moved all the way right for timer program to operate.

To check operation, manually rotate dial clockwise to watch lights turn on-off.
This will tell you exactly how to set pins and switch.
To set current time, rotate dial clockwise until current time lines up with arrowhead located on inner dial.
Dial rotates as time passes, and pins trip a lever that opens and closes switch to your lights.

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GE 15070 timer does not have ground wire.
I assume this timer is double insulated, just like some power tools and electronics only have two prong plug.
Leave bare ground wire attached to other bare grounds in the box, and shove grounds to back of box.
I have one of these timers installed.
Timer has red, black and white wires
http://waterheatertimer.org/GE-timers-and-manuals.html#15070

How to wire GE 15070 timer:
1) Switch has 2 insulated wires. Take switch wires and connect them to timer-black-wire and timer-red-wire. Choose either one for now.
2) In back of electric box are white wires twisted together and covered with wire nut. Connect timer-white to these wires.
3) If electric box does not have white wires described above, then connect timer-white to bare ground, along with timer-green wire.
4) Turn on electricity, and slide switch to ON-or-OFF. If lights come on, timer is wired correctly. If lights do not come on, then reverse timer-red and timer-black wires.

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I believe your timer must be exposed to cold.
The CR2 battery in ST01C will only last 30 days when exposed to cold temperatures.
I was reading a pdf chart few months ago about timer batteries, and the power drop-off becomes really steep after 30 days.

Most digital programmable timers are susceptible to cold if they have battery or internal rechargeable battery that supports timer functions or acts as program back-up.
For example, I e-mailed Honeywell about their Aube rechargeable-battery-type-timer, and they said anything below 32F and the display becomes slow, and timer loses accuracy.

Your ST01C wiring is correct. I test timers. They don't operate if wiring is wrong.

I solved same problem with my back porch timer using GE 15070 mechanical dial timer. Home Depot sells in-store for $18.
Link below gives you photo, manual, and wiring instruction so you don't need electrician.
http://waterheatertimer.org/GE-timers-and-manuals.html#15070
Open page and notice the timer below the 15070 has a photo insert showing on-off pins located on timer dial.
You move pins with ink pen or fingernail file.
Of course this is not self-adjusting digital timer, so I just set it for times I want, and it's easy to change times. It will be perfect for 6pm to 11pm daily.

And then use ST01C on a indoor switch.
You can wire the ST01C yourself by simply looking how each wire is connected now, or you can add a comment to this thread later and I will answer:
http://waterheatertimer.org/ST01C-program.html#ST01C

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15086 timer:
Pressing door will override timer setting and manually turn switch ON-OFF.
Press door or open timer door, and press any of 4 buttons located in corners.
You know information posted so far:

However, you are moving the small switch from Timer to OFF.
And when switch is OFF, the override does not work because circuit is OFF.
You want a timer that can switch programming OFF, and not switch circuit OFF, so that device toggles between timer and ordinary switch.

Looking at timers on the market today, I cannot think of any in-wall programmable or box timer that does what you want. Each has manual override that reverses timer function, but when you set timer to OFF, the circuit is OFF.
http://waterheatertimer.org/Programmable.html#Aube

Plug-in-type timers have override button that turns programming off, and then you can operate table lamp normally with the lamp switch. But in that case, the lamp has separate switch that operates outside of timer functions.

GE 15070 24-hour mechanical timer could do what you want:
http://waterheatertimer.org/images/GE-15070-mechanical-timer.jpg
In this specific case, when you wanted to operate timer like a light switch, you would move all the pins inward, and then use the on-off switch. The problem with this idea is the pins are tiny, and you need very good eyes or a light-source and small tool to continually adjust pins.

See GE timers and manuals:
http://waterheatertimer.org/GE-timers-and-manuals.html#15086

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The Grasslin analog timer works same as GE 15070
http://waterheatertimer.org/GE-timers-and-manuals.html#15070

You can also download Grasslin manual:
http://www.intermatic.com/products/grasslin/electromechanical%20timers%20and%20timer%20modules/in-wall%20timers.aspx