I don't see any day as better or worse than any other day. The only number that should really change is the prepaid interest amount due at closing. In a refinance, if you fund on the 10th, you will have 10 days of interest to pay from your old loan and the remaining 20 days of the month will be interest due on the new loan. Either way, you are paying 30 days worth of interest.
Some mortgage companies may impress upon you the importance of closing prior to month end. Many times that is because they get paid a higher split for a higher loan volume in each monthly period.
