'availingly' ultimately derives from the verb 'avail', which comes from Middle English 'availlen' (or 'availle') and Old French 'availer', ultimately from Latin 'valēre' where the root 'val-' meant 'to be strong, to be worth'.

'valēre' (Latin) > Old French 'availer' > Middle English 'availlen/availle' > modern English 'avail' + adverbial suffix '-ingly' (formed analogically; usage rare/archaic) to produce 'availingly'.

Initially related to the basic sense 'to be of strength or worth / to help', it evolved into the idea of producing benefit or usefulness; the adverbial form came to mean 'in a useful or advantageous way' (now rare/archaic).