March 29th is the Friday before the 1st of April, the new financial year. That gives financial institutions the weekend to calm their nerves, get IT systems up and running and make a "clean" break on the first day of the new financial/tax year.
Of course the plan is to have a transition period during which nothing much changes, so this may be an unimportant date for financial institutions with the projected end of the transition on December 31st 2021 being the big change-over.
Then again, if there is a no-deal brexit the 29th of March will be the cliff edge and the government will use the weekend to try to prevent Monday being a huge financial crash.
Arguably 9 days was not enough, the government might have done better to make it a year and 9 days to give itself time to prepare a negotiation strategy, but the weekend falling just before April 1st seemed to present an opportunity.