I have some experience in this area.
What type of current are you trying to measure, AC or DC? How much current? What voltage level? How many points?
I'm partial to the Arduino Mini Pro, it's small, draws very little power and is available in 5V (5-20V supply) and 3.3V models. You can buy them on ebay (or direct from China) for about $2 each.
For something with a bit more horsepower I really like the NanoPi Neo, it has a quad-core Arm processor that runs at up to 1.2 GHZ and draws about 1 watt of power. I'm using one right now as a combination Web Server, Email Server, DNS, file and media server (music an
video), as well as the central data collector for my network of sensors. With 512MB ram, the optional heatsink (highly recommended), and shipping it costs about $20.
Or you can buy it with an aluminum case and OLED display for about $40 delivered.
The advantage of having a server online constantly is that you can connect to it at any time using a smartphone, tablet, computer, etc. and check the status right then. This might allow you to catch a problem before it develops into a failure.
This is an example of part of my power monitoring system display: