I’d give her three weapons: one ranged, two close combat. I’m basing my choices below on her background and I’ll explain why.
These three cover your basic necessities while giving you the ability to branch out over the course of the story. Let me explain each in detail:
The Sling – the sling is the weapon of the farm child, one children still use to defend their flock from attackers both animal and human. It’s a great weapon, easy to maintain and use with ammunition everywhere with the potential for deadly accuracy.
Like the sword, the sling comes with it’s own thematic history when found literature as it was the weapon of choice for David when he fought and killed Goliath. It is a literary flashcard that this person is an underdog meant for great things.
Yes, it’s the weapon of the child but your character starts the story young. This is their coming of age. Besides that, it’s a great weapon and one you don’t have to worry too much about when it comes to logistics. The only downside is that its themes indicate someone who is intelligent, canny, and cunning, who performs the unexpected, and breaks the conventional rules to change to odds to their favor. (It also takes way less time to set up than the bow.)
The Staff – the staff is a basic, easy to use weapon that has an advantage over the sword in terms of reach and provides the base for training in more advanced polearms like the spear or the halberd. The learning curve is quick, it’s easy to practice, and it’s the weapon of the wizard/traveler.
If your character is going on a journey, she needs a walking stick. A walking stick that stealthily transitions into a beat down stick. She’s not professionally trained so taking on a city guard or knight early in the story is going to be a last resort, but she couldn’t have done that with an axe or a flail/mace anyway. If asked by the local city guard or lord, she has a convenient justification for carrying it and given that she is a peasant this is probably a good thing.
The Dagger – the dagger covers basic hand to hand, if she can’t get to the staff or the sling, then she can fall back to this. The dagger will give her the advantage in an unarmed/unarmored fight which she needs because she doesn’t know how. She may need someone to teach her how to hold it and strike with it, but it’s another weapon that is very easy to learn the basics of.
Other suggestions: a club, a cudgel, or a hand axe (like a hatchet)
On the mace/flail/morningstar: these are weapons meant for armored melee, specifically against enemies in plate. They won’t do your character a lot of good if you’re not planning on having her go after guys in mass melee wearing heavy plate. Since she hasn’t been trained to use it or fight them anyway, I’d suggest avoiding it.
Between the three, she has better odds than if you give her just the one. Also, the one weapon concept is really, really stupid. She’s going to face a variety of challenges over the course of her journey and there is no one weapon fits all. Different tools for different challenges. This is necessary to understand if you ever want to break out into more specialized weaponry because it’s important to remember that a highly specialized weapon only has an advantage under a very specific set of circumstances and combat variables. Any of the above weapons (ignoring the flail, the mace, and the morningstar) are weapons that would be ones she’s grown up with and make sense given her background.
There are some important things to remember: The weapons I’ve suggested won’t help her when it comes time to take on a professional warrior a la a knight but given your character’s background there’s no real helping that anyway. The axe won’t help her either. They will help with encountering bandits on the road during her travels and defending herself in tavern brawls, weapons that will provide her with the opportunity to fend someone off and give her time to adjust to her new surroundings. Some weapons like the sling are preemptive and can be used dangerously when at range to take out men in armor if they’re not wearing helmets. It won’t help her if they see her and catch her.
Combat is about skill and experience than it is about any physical qualities. Not all weapons are created equal and it’s best not to underestimate the training of the enemies she will be facing. It’s best to remember that these guys are dangerous to her and direct force like taking on a whole garrison full of guards to get to one single target may not be the wisest choice. (Is it ever?) Despite their lack of experience, your character has a unique perspective to apply when solving situations. Let them live in that place. Little John wasn’t a super fighter but he still managed to dump Robin Hood in a river.
A hero is made a hero by their brains, not their brawn. By understanding their limitations, you’ll be better able to work out how they might uniquely solve their problems. Fighting a better equipped enemy on the enemy’s terms is not cowardice, it’s stupid. Your character will never be able to “catch up” to other characters that have been training to become warriors since childhood, just like how during the Star Wars Original Trilogy Luke never really became Vader’s equal in martial skill. He matches him in other more important qualities and those qualities are what cement him as a hero.