The day of the month that you close on the purchase of your home is important and should be part of your contract negotiations. The amount of prorated interest that you will pay at closing will be determined by the day of the month you close.
Mortgage interest is paid in arrears. A later date in the month means less interest paid as part of your closing costs. For example, if you were to close on May 30, then you would only pay two days of interest plus the interest due for June. Your first payment wouldn't be due until July 1.
Since 95% of closings occur at the end of the month, and if a few hundred dollars at closing doesn't make a big difference for you, then you may choose to close earlier in the month, when title and escrow companies are not as busy and closings tend to go a bit more smoothly.