You can invite the person through the chat system or by leaving a comment. Note that a user needs at least 20 reputation to be able to chat, so check that first.
Note: Stack Exchange has three different chat sites -- one for Stack Overflow (chat.stackoverflow.com), one for Meta Stack Exchange (chat.meta.stackexchange.com), and one for everybody else (chat.stackexchange.com). In the following explanation my links are for the last, so if you're using one of the others you'll need to adjust that.
First, go into chat and enter the room you created. (This is important; you can't invite someone to a room you're not in.)
Next you need to find the user's chat profile. If you don't have a link to the user's chat profile, follow step 1 in this answer to obtain it from one's main site profile.
Let's suppose I want to invite you. On the chat user page you'll see several buttons, one of which is "invite this user to" and a drop-down list of all chat rooms that you're in.
Select the chat room and click the button; the user will be pinged. Note that it can take 20-30 minutes for the ping to arrive, depending on how recently the user has been in chat.
Then monitor your room and see if they respond.
I have a link to the user's chat profile but don't see an option to invite, either to the specific room or at all. Why?
First, confirm that you've joined the chat room to which you're trying to issue an invitation. As stated before, you can't issue invitations for rooms which you aren't currently in. You also can't issue a chat invitation if the user doesn't have write access to the room (if it's a gallery or private room).
If you still don't see the option to invite, there are three possibilities:
- The user has already been invited to the room. Only one pending chat invitation per room can be issued to a given user, altogether from all users. This means that if you or someone else invited the user to the chat room at any point in the past and they didn't respond to the invitation, you cannot invite them again. Chat invitations do not expire.
- The user has added you to their ignored users list.
- You are banned from issuing invitations or creating new chat rooms. This happens if you get kick-muted from a chat room three times in a row, for 30 minutes. To confirm if this is the case, go to
/rooms/new on the chat server you're trying to issue an invitation on; if you get an error message that you're banned from creating rooms, this is the case. This restriction is indefinite until a moderator removes it.
If you can't use chat directly, you can leave a comment inviting him into your chat room. Create the room, enter it, get its URL, and then leave a comment addressed to the user and including that link. This is, by the way, more public; anybody who sees the comment might choose to come into your room. But almost nothing is really private on the SE network anyway (for good reasons); I'm just pointing this out so you won't be surprised if it happens.