Low power timer circuit for solar tracking

Some more info on the charger would be useful (any specs written on the casing, in the manual, etc)
Hopefully it's a switching charger.

Anyway, some rough calculations:

Battery Wh = 1.1Ah * 3.7V = ~4Wh Typical car battery Wh = 50Ah * 12V = 600Wh

You say the device operates for roughly 6 hours on a full charge, so the average current and power consumption is:

1.1Ah / 6 = 183mA 4Wh / 6 = 678mW

If we assume the battery is in good condition and can supply, say 300Wh (half of it's capacity) and still start (I'm no expert on car batteries so this is a bit if a guess, but I'm pretty sure less than 50% charge is a bad idea) then the operating time is:

300Wh / 0.678W = 442 hours or 18 days.

Now the above does not include the charger inefficiency and battery drain caused by other electronics in the car, so it seems feasible that you could halve this figure quite easily.

So although making sure the charger is doing a good job is certainly a good idea, I think it is likely that just shutting off the charger periodically may not be the answer, and you need to look at lowering the devices consumption (e.g. transmit data less frequently) and/or installing a larger capacity battery. Also, if the car is not in a garage, one of the readily available dashboard solar panels would help to keep the battery topped up.

Simple Timer Circuit

Since there has to be many such circuits already out there, rather than draw it from scratch I had a quick look around to find the kind of thing I had in mind that didn't involve a microcontroller (sorry it's a bit late, some urgent stuff came up)
Anyway, this circuit at http://www.electronics-project-design.com/electronictimerswitch.html seems to almost fit the requirements quite nicely, and is based on easily swappable and cheap components. It can be upgraded if necessary (e.g. the oscillator could be crystal based for better accuracy)

The IC is a 14-bit binary counter, and is clocked by the RC oscillstor made from pins 9, 10 and 11. The values of the RC components set the clock period (more info in the C4060B datasheet)
To set the on/off period, you need to AND the correct pins together for your desired ratio. Say you set the timing so the the count reaches 2^14 in 60 minutes, this means the clock is 2^14 / 3600 = ~4.55Hz.
Now you want the timer on for 7.5 minutes of this hour, so you need it on for 2^14 / 8 = 2048 counts. So to calculate the bits we need to AND together:

2^14 - (2^14/8) = 14336 to binary equals:

11100000000000

So bits 13, 12 and 11 need to be ANDed together (the same as the schematic above just without D1)
This is just a rough example, other timings can be achieved by calculating accordingly and you could use proper AND/OR gates, or a comparator IC or cascade another counter IC or... if you want to improve the design. Hope this helps to get you started.

As Nasha has already pointed out, using a CMOS 555 will greatly cut down on the quiescent current required from the battery, as well as eliminate the huge shoot-through current spikes inherent in the bipolar 555's output totem-pole every time it switches.

Another area of concern is the timing capacitor's chemistry and capacitance.

In order to minimize its own leakage, the cap should be something other than electrolytic, (plastic or ceramic dielectric, say,) and its capacitance should be selected to be as small as possible and, in the circuit you've linked to, the value of the resistors should be made as large as possible since every time the cap charges and discharges all that energy is just wasted.

Also, C2 can be eliminated.