Someone rightly pointed out there aren't any important festivals that we celebrate in the month of May. That is if you do not count Chithra Pournami as a festival celebrated at home. Chithra Pournami is the full moon day in the month of Chithirai (April-May) and is celebrated as the birthday of Chitragupta. This year Chithra Pournami is on the 9th May.
Chitragupta is the accounts keeper of Lord Yama. He keeps the accounts of the good and bad deeds committed by the humans and advises Lord Yama of the appropriate place for them in heaven or hell at the end of their lives. There are various stories about the birth of Chitragupta. Yama found it very difficult to keep track of the good and bad deeds committed by the humans and pleads with Lord Brahma to give him an assistant who can keep track of this. Lord Brahma goes into deep meditation and at the end of his meditation finds a young man in front of him with a pen in his hand and names him as Chitragupta and assigns him as the deputy of Yama.
In another story Chitragupta is depicted as being given life by Lord Shiva to a portrait drawn by Devi Parvathi. Since he emerged from a Chithiram (portrait) he was named Chitragupta.
It is also said that Indrani, Lord Indra's consort, wanted to have a child and prayed to Lord Shiva. Lord Shiva asks Chitragupta to be born as the son of Indrani. So meticulous is he in keeping the accounts of everybody's destiny that Chitragupta refuses to be born from the womb of Indrani as Indrani is not destined to have children. Lord Shiva then sends Kamadhenu, the celestial cow, to Indra's abode and asks Chitragupta to be born from the womb of Kamadhenu.
Chitrapournami is celebrated as a big festival in many temples. The Chitrapournami festival in Madurai is especaially famous.
Our Echiyamma (my beloved grandmother) used to celebrate (or perhaps appease Chitragupa) Chitrapournami by gifting a bamboo tray (Muram in tamil and malayalam) filled with fruits, idli, payasam and an iron ezhuthani (a metallic scriber with which inscriptions were made on palm leaves in the olden days) to a brahmachari (bachelor). She would tell the brahmachari to return the ezhuthani later and would pay him four annas in lieu. The ezhuthani would be kept safe for use for the coming year. So for us Chitrapournami was also a festival at home since we got a feast on that day.
Though there are not any festivals during this month, in Tamil Nadu and Kerala this is the time for the festivals in various temples. In Kerala, the festival season starts by February at the end of the harvest season and goes on till the end of May or upto mid June. Of these the Thrissur Pooram festival is the biggest event of all and is a treat to eyes and ears.