My 5 year is a more reticent fellow by nature but even he cycles in and out of bouts of deeper shyness. We ignore the shyness. We believe his shyness is a result of his greater awareness of the wider world but also just who he is. He takes things in and that reads as shy.
We can see him developing a sense of awareness of people’s judgments and opinions, which leaves him feeling vulnerable and/or exposed to ridicule. We think he is wrestling with bigger concepts which can be difficult to articulate and work through. For example my husband and I have slipped up and made negative comments about others which my son has overheard. Sometimes you can see his wheels spinning and then see the resultant shyness emerge. In these instances we tackle it sideways with regular conversations. We the adults remind each other to mind little ears and to be less negative in general to set the proper example. If we catch ourselves in the conversation, we will also apologize to our son and to each other. Reminding everyone to be kind to others because being mean is easy but well, mean. We also talk to our son about people’s feelings, how people have opinions and how a person can feel about someone else’s opinions. It winds up being a running series of conversations circling around remember how words can hurt, how opinions can hurt, minding other people’s feelings, not retreating from all people over one mean person, etc. The harder conversations stem from when one of his little friend’s makes fun of him over something. Recently we had one of them tell him how only girls like such and such TV show. We talked to him about how variety in people, opinions, dress, play, toys, everything is a good, positive thing. We also explained there aren’t girl or boy things so why can’t he enjoy those TV shows? Honestly we have had this conversation more than once because my son tends to the “feminine” as defined by stereotypical “little boy” standards.
In general we opt for the ignore and observe approach to avoid blowing up nonissues or jumping in to too hastily. Good luck.