If your daughter is 9 (which I see in others' answers but not your post), she is perfectly capable of doing what needs to be done even if it isn't what she would choose. She goes to school and functions from 8 to 3, more or less, right? She might be better in math than the next kid, but a 3rd kid might be better at reading than your daughter is. Doesn't she function? Aren't there other kids in lower grades in her school with whom she shares the playground, the lunchroom, the hallway, the bus ride? So she's quite capable of being in a room with a 6 or 7 year old, with both of them making some adjustments to the other kid.
I wonder if she's used to "just the two of us" and doesn't like to share? That's understandable, but you are her mom and not her best friend. She needs to learn that you call the shots, not her. And she doesn't get to be rude or, frankly, ruin your visit just because she's bored. Ideally, she would find something fun to do with the other kid and be polite. Surely 2 kids can play Connect 4 or Go Fish even if they are a few years apart. It's okay if they do their separate activities too, although it's a lot less social/polite. Besides, if you should start dating again and have someone serious in your life, what's the plan? Let your child decide whether to pitch a fit or just look miserable? You'll regret that.
I think, the next time, you have a sit-down with her beforehand. I'd even discuss it with her now, while this visit is fresh in her mind. Tell her that she does not get to ruin your visit with a longtime friend because she thinks you are her entertainment system. I would tell her she is expected to come in and be polite/decent. She should pick one activity to bring with her and say to the host child, "I brought ____, would you like to play it with me? Or is there something else you have that we could both do?" She's taking the lead a little as the older child with leadership qualities, but also being deferential to the host. She needs to learn these skills for entertaining in your home as well - making others feel welcome and not only doing what she wants to. Show her how to do this - when you have people over, discuss out loud what you are thinking of serving or cooking because "I know they like it" or "I want to offer a few things so I'm sure everyone will be able to enjoy something."
I'd also have her take ownership of the "back-up activities" so you aren't in charge of everything and she doesn't get the right to "shoot down" every idea. She's 9, not 3. She can step up her game here. Otherwise, you can tell her that anyone too little to be polite and entertain herself is too little to ______ (play on a computer, stay up past 8, have a sleepover, get an allowance...whatever Big Kid things she wants to do). There is no way I would have left after 1 hour just because a 9 year old was pouting. She could sit in the car for all I care, but I'd be having my visit. And I wouldn't make excuses for her. I'd just say to my friend, "Petunia needs some time to herself to find her manners so she's going to the car to cool down for a while and be with her thoughts."
You do need to have some babysitting options. Maybe it's trading off with a friend who can't afford or find a sitter either. And the kids just learn to adapt. I also always recommend some good library books on a particular subject. Sometimes talking to the children's librarian about recommendations either at your library or on inter-library loan will turn up some wonderful examples to read about (rather than you just lecturing her about). Librarians are wonderful resources and they love to do a lot more than just help people scan their library cards!