Usually the first thing to delegate are props.
Working props should be sourced by the ASM, and the ASM should do pre-rehearsal checks to make sure all props are ready for the scenes being rehearsed that day.
When it comes to the show, the ASM should manage the props table - ie make sure everything is preset and ensure that everything is returned at the end of the run.
Next, I would get the ASM to help with any understudy rehearsals.
Third, I would ask her/him/it to attend any meetings you have with LX/Sound/Set/Whatever - even though those meetings may not effect her/him/it directly, it is good to have someone else who is kept in the loop.
I think the key to keeping an ASM keen is to define a few jobs that are theirs alone - so it may be a matter of sitting down with them and discussing your duties and what areas you think they can look after, then asking if there is a certain area they would like to focus on. If your ASM wants to move to an SM sometime in the near future, you may want to arrange that they maintain their own blocking script and you can compare notes after rehearsals, or maybe give them the job of prompting - leaving you free to focus on blocking and other issues.
During the run, apart from managing props, maybe give them the task of ensuring all actors are in position for their entrances. If she is new to ASMing - as far as paperwork goes, entrance/exit charts are a good 'toe dipping' exercise. Add to that that the ASM should either lead the deck crew or if you have a deck master - should be assisting, I think you will find that more than enough to keep him/her/it busy during the run.