To increase the time you have the lamps will light at night you have to add a supplemental battery to store enough charge to run the lights at night after the batteries in the individual lamps would have run down to a point that they need help, they need a little boosting power at night.

So you need three things: A buck converter to convert your wallwart power to the voltage of a reserve battery. You can make the battery with individual cells to make a battery from 3 to about 12V.

Now you need to charge the reserve battery. This is where the either a simple single IC voltage regulator or a buck converter would be most efficient to do this job using the wallwart, if required during the day. The solar cell may be able to do most of the job, with the wallwart taking up the slack.

You can buy a dry lead-acid battery and just float charge it with a simple single regulator IC, a 4-component LM317 regulator to set the maximum "float" charging voltage to approx 13.6 Volts. You can charge this lead-acid cell with your solar cells using a boost converter, if enough sunlight is available.

If your 16V wallwart is efficient, there may be little or no need to turn it off when charging is complete, since the charging circuit will basic draw very little power.

Next, you need to convert the energy in the reserve battery to a voltage compatible with each of the individual lamps. You need to convert the reserve battery voltage down to 1.7V.

For this you will need an efficient buck regulator to convert the reserve battery voltage, which might be anywhere in the range from 3 to 13.6V to the 1.7 Volts needed to supply the lamps at night.

A simple one IC switcher module can be easily built on breadboard or purchased on ebay for just a few bucks that will do this step-down job well and reliably.

If you don't want to use a 12V lead-acid battery, you can still use the wall wart with a step down buck converter to charge a two or three Li-Ion 16850 cells in parallel to 4V of charge or else make a battery >= 3V out of a number of series-parallel connected rechargeable NiMH cells. You can attempt to charge this reserve battery with the output of the solar cell with a boost converter.

You can use a boost converter to convert the output of the solar cell to charge the reserve battery instead of the wallwart, but, as necessary, switch to the wallwart if the charge is not complete by late afternoon or cloudy days.

At some time in the afternoon, if the reserve battery is not fully charged, then you use the wallwart, You turn it on with a timer/relay to bring the reserve battery up to full charge by the above stated methods.

In this way, the solar cell and the lamps' individual batteries will do their best to supply energy to the reserve battery and lamps during the night while the reserve battery with a buck converter will supplement the lamp batteries with steady 1.7V power to keep things lit all night long.

To save the maximal amount of energy by this scheme, a simple Arduino or PIC MCU circuit could be used to manage the timing of this circus, getting feedback from a photo sensitive device to sense sunlight and A2D circuits to measure voltages and timers to and turn on/off the wallwart as required. It is really a quite simple program to write even for a beginner using an Arduino.

If programming a MCU seems to intimidate you too much, a simple mechanical timer can turn on and off the wallwart to charge the reserve battery, so then you only have to buy a step down buck converter and a step up boost converter from ebay.

 

Here is a plot from Eveready that pretty well sums it up. Some thing to note:
1- Fully charged voltage is 1.4 volts. This drops quickly to 1.2. So 1.25 won't trickle charge it.
2- Under no load the 1.7 might, but you solar cell is also probably about 1.7 volts in bright sun. You might be able to tell if there are 2 or 3 individual cells by looking at them. The open circuit voltage is about .58 volts per cell.
3- From the graph it is dead at .85 volts and its resistance should be around .1 ohms. This was my concern about the charge current for 10 of them when they were dead, but your measurements put that to rest.
4- I'm not sure why you are worried about the 5ma. When all ten are hooked up you have a 9000 mah capacity so it only about 0.5% of your capacity.
5- the 317 likes some load there that the 240 ohm supplies. However having said that, you could probably make it bigger without actually hurting anything.
So my take is any of them should work but option one gives you the best battery life. But even that doesn't matter because you could remove the batteries if they go bad and just plug it in.

 

Here is a plot from Eveready that pretty well sums it up. Some thing to note:
1- Fully charged voltage is 1.4 volts. This drops quickly to 1.2. So 1.25 won't trickle charge it.
2- Under no load the 1.7 might, but you solar cell is also probably about 1.7 volts in bright sun. You might be able to tell if there are 2 or 3 individual cells by looking at them. The open circuit voltage is about .58 volts per cell.
3- From the graph it is dead at .85 volts and its resistance should be around .1 ohms. This was my concern about the charge current for 10 of them when they were dead, but your measurements put that to rest.
4- I'm not sure why you are worried about the 5ma. When all ten are hooked up you have a 9000 mah capacity so it only about 0.5% of your capacity.
5- the 317 likes some load there that the 240 ohm supplies. However having said that, you could probably make it bigger without actually hurting anything.
So my take is any of them should work but option one gives you the best battery life. But even that doesn't matter because you could remove the batteries if they go bad and just plug it in.

Ronv:
1) Can you send me the link to the eveready voltage graph. I tried to find it on the internet.
2) Do you think I require a 'larger' heat sink on the LM317?
3) Is there significant concern on 'static elec' when soldering the LM317 with an ordinary soldering gun?

 

Sure.
1- http://data.energizer.com/PDFs/nickelmetalhydride_appman.pdf
2- Bigger would be better.
3- I wouldn't worry.

 

Sure.
1- http://data.energizer.com/PDFs/nickelmetalhydride_appman.pdf
2- Bigger would be better.
3- I wouldn't worry.

Ronv:
Thanks for the link to data.energizer.

I transfered my components from breadboard to circuit board (no diode installed). I added a 150 ohm and red LED between Vin and ground. It looks like maybe some connections on prototype were not that good because my results are different. I am now getting the full 1.23 v at the lights and sufficient ma to run the 2 lights when batteries are low so I am keeping my 1K pot at the adjust to zero.
Is my 150R and LED addition OK?
My measurements with 2 lights on with no batteries in and ampmeters at Vin and Vout and 1K pot at zero:
1) Vin 5 volts, 140 mA
2) Vout 1.23 volts, 110 mA
3) Vout with no load and 1K pot set to max: 4 volts (this is just for your info)
Doing a 'power' balance:
Pin= VinIin = 5 x .14 = 0.7 watt
LED: 2.5V x .017 = 0.04 watt
150R: 2.5V x .017 = 0.04 watt
Pout: 1.23 x .11 = 0.135 watt
Balance: 0.7 - .04 - .04 - .135 = 0.485 or 1/2 watt
Is my math roughly correct?
Where is the remainder 0.485 watt being consumed? Are my 2 multimeters that are measuring the current consuming a lot of this watts? Is this reasonable approximately for this circuit?