I realized in November while reviewing my compassion.com account that I only imagined having sent a birthday present to my sponsored child for his August birthday. So, I sent it in November, it will arrive January/February (now), and I am confident that my child will enjoy it now. I did write a letter mentioning that I was profusely sorry that I forgot (I wrote him a letter for his birthday). I sponsor two kids at the same project, and the main reason I specified it as a birthday gift was to try to avoid any appearance of playing favorites as I remembered the other child's birthday.
If I only sponsored one kid, then I would have sent the when I remembered as a general child gift, which is another gift option. Many children do not get birthday gifts, so don't feel bad if you cannot send one. I would write a letter telling your kid happy birthday whether you are able to send a financial gift or not. Sending a picture or postcards with a letter is a tangible gift they will enjoy even if it is not financial.