new_variable=$( awk 'BEGIN{RS=ORS=" "}!a[$0]++' <<<$variable );
Here's how it works:
RS (Input Record Separator) is set to a white space so that it treats each fruit in $variable as a record instead of a field. The non-sorting unique magic happens with !a[$0]++. Since awk supports associative arrays, it uses the current record ($0) as the key to the array a[]. If that key has not been seen before, a[$0] evaluates to '0' (awk's default value for unset indices) which is then negated to return TRUE. I then exploit the fact that awk will default to 'print $0' if an expression returns TRUE and no '{ commands }' are given. Finally, a[$0] is then incremented such that this key can no longer return TRUE and thus repeat values are never printed. ORS (Output Record Separator) is set to a space as well to mimic the input format.
A less terse version of this command which produces the same output would be the following:
awk 'BEGIN{RS=ORS=" "}{ if (a[$0] == 0){ a[$0] += 1; print $0}}'
Gotta love awk =)
EDIT
If you needed to do this in pure Bash 2.1+, I would suggest this:
#!/bin/bash
variable="apple lemon papaya avocado lemon grapes papaya apple avocado mango banana"
temp="$variable"
new_variable="${temp%% *}"
while [[ "$temp" != ${new_variable##* } ]]; do
temp=${temp//${temp%% *} /}
new_variable="$new_variable ${temp%% *}"
done
echo $new_variable;