Odd Behaviour of Arduino

I think I have got too used to working on large computers, which have essentially infinite resources, as far as my little projects are concerned, so I am having a bit of trouble with Arduino.   The two interesting problems I have seen are:

1. Low Battery:  The symptoms were the device operating ok for quite a while (initially around an hour, but later ~5 minutes), then doing very unexpected things - what I saw was the pin 13 LED flashing on and off, but my software was not doing anything with that pin.   It looked like the board had just lost its marbles.   It turned out that the issue was low voltage on the 9V battery that was powering it - I did not realise initially because the LED back light on the display worked fine, which is the test I was using for 'it has got power'.  When I put a volt meter on it, it was only providing around 5.3V, so I think that this was insufficient to start the arduino properly...but surprisingly was enough to light the LED ok - I have always thought that LEDs need more power than little electronic devices, so if the LED is ok, it has enough power - this is not true, so I must think of another simple check!
2. Out of Memory:  My solar thermal monitor now works nicely with an LCD display, and I made a simple test program to write data to an SD card.  The odd thing is that merging the two together results in a program that compiles ok and loads onto the device, but when it tries to write to the SD card, the arduino re-starts (I had worse effects earlier when it would just not start at all, until I removed some code in the setup function that writes to the SD card).  I think it must be running out of memory, but I need to do some work to check this...will update this once I have fixed it.