not really sure if I would do anything without meeting the family and child. What if you took the holding fee and had someone great show up at your door step for care the very next day but had to inform this person in the flesh that you are holding a spot for someone youve never met. I wouldn't take anything until meeting them face to face.
Personally I always keep advertising and interviewing until someone has signed a contract ~ so while I would allow the mother to book a interview at their earliest convenience I would make it clear that the first person to 'commit' will get the spot so if you are truly interested in securing daycare than arrange to get to town ASAP and tour some programs and commit to one but I would not be interviewing the grandma or making any commitments to them with a holding contract without meeting the ACTUAL clients cause there are just too many ways that can backfire!
good points.

Its not like my phone is ringing off the hook, but hopefully that will change soon.

If they really want the spot they will find a way to get here next weekend.