As far WordPress is concerned – your multi-dimensional array is one option. To update just part of the multi-dimensional array its necessary to retrieve the entire array, alter it accordingly and then update the entire array.
Suppose your multi-dimensional array is as follows:
my_options = array(
'foo' => 'bar',
'hello' => 'world',
And suppose you want to update the value of the ‘hello’ option from ‘world’ to ‘moon’
//Get entire array
$my_options = get_option('my_options');
//Alter the options array appropriately
$my_options['inner_array']['hello'] = 'moon';
//Update entire array
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