As a non parent with no experience in this, I will give my two cents. As a disclaimer, there is no objective answer, the entire situation is subjective to how you feel. With that said, my answer is this:
TLDR; When you start feeling comfortable enough about this person to make a serious commitment to the relationship.
The biggest trouble for your child with introducing them to someone is them getting an short-lived attachment to that person, and having it not work out. The Huffington post has a great article on this exact issue, and makes good points from both perspectives (parent vs partner). The seriousness of the relationship should really be the main determining factor* in deciding whether or not you should introduce them to your child.
“Introduce them as soon as you even start to consider getting serious. The interaction with the kids will help you decide if he/she is worth another step toward commitment. That doesn’t mean overnights or PDAs, just ‘friend’ behavior. If your potential partner can’t respect that, or they are lousy with kids, then kick them to the curb. You are a parent first.”
Your child is 3 years old, prime age for beginning to form opinions on any and every topic. If you think the relationship is serious enough to last to the "this could be my child's future parent" stage, don't suddenly invite them to live with you, but rather in drips and spurts. Your child will need time as well to form their own opinion on your partner, so provide a forum to do so. Family activities like going to the park for a few hours, having a movie night, or a family dinner are perfect opportunities for the three(or more) of you to spend time together.
*The age of your child may affect when you want to introduce them in general as well. It's a much different story for a 3 year old and a 13 year old to meet a prospective step-parent. If your children are emotionally mature enough to understand what having a step-parent entails, you may be able to introduce them to your partner more easily and quickly, as they will know that a long-term relationship may not be on the books for this person. On the other side of the coin, if your child is much younger (as is your case), you might want to wait a bit longer to introduce them; as the attachment may come more readily, not necessarily for the better.
“It depends on the age of the kids. I felt that it was important for my teenaged daughters to watch me go through the dating and selection process.”
"Saddling kids with attachment issues of their own while their parents are looking for ‘the one’ is a bad idea. At the same time, kids should be involved in any decision that will affect them; to tell your kids, ‘we’re engaged,’ when they don’t even know the prospective spouse is just cruel."
Don't wait too long to determine the seriousness of the relationship either. As a single parent, you can't really afford to hedge your bets on a "maybe" relationship. Decide for yourself whether it will work out or it won't, and be sure to stick to your decision. While deciding if it will or won't, you'll want to have a long term discussion about your goals for the future and where you see each other in it. It's imperative that you know what their ultimate goal with you is, and how it will affect your relationship with your child. Keep in mind not to wait too long, for they could be horrible with children
“Don’t wait too long to introduce the kids. What if you get emotionally involved with someone, wait months for him to meet your child, and then come to find out the man has no interest in your child after they meet? You’ve just wasted months of your life and your emotional investment in a person.”
My advice also would never be to introduce anyone you're not exclusive with. If you're still seeing the other parent of the child on a regular basis, they may have something to say about your choice of partner and may cause friction in the relationship that you wouldn't want to expose your child to.
If it’s possible, have the conversation with your Ex letting him know you are about to introduce the children to your new beau and you would like for him to meet him first. If your Ex is accepting and welcoming the children will be as well. You can make them feel even more assured and mention “your dad met Mike last week” and we all have your best interest at heart. This will make the children feel relaxed and remove the pressure of them thinking they have to choose sides.
As a last note, don't be afraid to be affectionate (within normalcy) with your partner after they have met. This will send the message to your child that "Wow, this new person loves my parent a lot, and will be less likely to hurt them" and will help to show your child that all parties want to make the relationship work. Your child loves you just like you love them, and will want the best for you regardless of the situation (again, within normalcy). From another article:
Let’s face it, one of the biggest concerns from your child’s point of view is, how does he treat my mom? So if you are giving and receiving hugs or holding hands only when your child is out of sight, they don’t get to see what you see. And since we are setting examples for our kids for their relationships when they get older, show them that it’s ok. A warm hug cures all, even for grownups. We encourage affection to the children as well. They need to know that he not only loves mom but he loves the kids too.
In conclusion, there's no set-in-stone time to introduce your child to a significant other, but by taking the steps to make sure the relationship is a serious one and that your child is protected from unwanted emotional harm in the process, you can determine when the right time to introduce someone to your child is. With the right steps and reassurances, only you are in the best position to determine if the time is right for your child to meet a new adult.