I am shocked to hear that some people believe $5 an hour is acceptable for childcare - you can't get a teenage babysitter around here for less than $8 - $10 per hour. IMHO the op is getting a GREAT deal - you can't put a price tag on piece mind knowing a trusted family member is watching your child.
I think that "daycare" and "babysitting" are two different things.
For a babysitter... come to my house, focus only on watching my kids... because I want a night out or whatever... $10/hr is appropriate. Not to metion, this is occasional, not a set schedule, and for short durations (a few hours now and then)
Daycare is a whole nother deal. Either the provider is taking care of more than one kid (everyone shares the load) or is like OP's cousin, running errands, living life with child in tow.
Consider, you have a baby and work full time (8hours+ 1 hour commute) that $10/hr you think is reasonable is $90/day... $450/week... $1700 mo! Yes $1700/mo for one child, full time if at $10/hr. Imagine you are an average family with 2kids and you put them in $10/hr daycare. That would be $3400/mo!! Would you still find that reasonable?? How would you even afford to work??
In my area (suburban Houston) fulltime childcare runs $700-$900/mo... yes about half what you "pay a teenager to babysit"
Effectively that is, $5/hour... as I say, either the provider is charging 5 kids that rate (making $25 an hour!) or running around doing much of the same things she would do, kid in tow or not.
Now, would I ever dream of going out for 4 fours on a Saturday night and only pay $20? No, but that's different...that's BABYSITTING