Well, I just helped a friend name a similar business and we used the initials of her family members to come up with a name.
Since this is a new type of business, I think the name or the slogan should describe what you do. Something like "Vonya's Personal Errand Service" tells what you do and it's personal, too.
With that, you don't need a slogan, per se, but could add one if you like, or could change it at will (seasonally, for example--"Your Santa's Helper" at Christmas).
For general purposes, I'd suggest focusing on the main benefits you'll offer--saving time, decreasing stress, etc.
"We run your errands so you can live your life"
Hope this helps & good luck with the new business!