The preposition, 'by' is used along with a date to indicate that a certain event or activity happens on or before the said date. The submission of names (or 'giving of names') has to be done on or before 15th May. Hence, the appropriate preposition which can be used in the blank no. 29 is 'by'. Hence, option A is correct.
The preposition, 'until' when used with a date/time means 'up to (the point in time or the event mentioned)'. It is not, however, used along with the auxiliary, 'should' + verb. It is used along with the auxiliary, 'can'+verb. The phrase, 'can give their names until 15th May' is correct. However, the phrase 'should give their names until 15th May' is incorrect. Hence, option B is incorrect.
The preposition, 'unless' means, ' except if'. It is used to introduce the only circumstances in which an event can happen . Eg: 'Unless you complete tour homework, you cannot go down to play'. Given this meaning and usage, the word, 'unless' cannot be used before a date. Hence, option C is incorrect.
The preposition, 'of' is used to show possession or belonging. Given this usage, the word, 'of' cannot be used before a date. Hence, option D is incorrect.