Best way to make variables local in a source'd bash script?

Wrap the whole script you want to source into a function, add local before the declarations you want to only use in the function, and call the function at the end of the script.

func () {
    local name="My awesome job"

    nowTime=`expr $(date +%s) `
    lastActiveTime=`expr $(date +%s -r ~/blah.log)`
    local secsSinceActive=`expr $nowTime - $lastActiveTime`

    currentRaw=$(cat ~/blah.log | grep "Progress" | tail -n 1)
    if [ -z "$currentRaw" ]; then
      local statusText="Not running"
    else
      local statusText="In progress"
    fi
}
func