Accept her challenge, and offer a wager

Assuming you chose to ignore her, odds are she'd complain about it the next time you met. You didn't really want to battle her at all, but on a practical note it would help you and Eevee learn how to work together against other Trainers. What finally convinces you to battle with her though is the chance to get something more than just experience out of the match.

You agree to her demand for a Pokemon battle and head outside, much to her grandfather's relief you imagine. Before the battle begins though, you ask her if she's willing to make a wager on the outcome of the battle. At first she dismisses your offer, assuming that her victory is already decided and thus has no need to bet on the outcome of the battle. Luckily, you know which buttons to push and from that point its easy to settle the terms of your wager with Blue.

From there, the battle starts between her Dratini and your Eevee. Blue commands Dratini to use Wrap. Knowing how dangerous that attack could turn up, you tell Eevee to dodge. Eevee is able to evade Dratini's lunge, leaving it open to a Tackle attack. Dratini is thrown back a good distance, but the battle is far from over.

Most of the battle is spent with Eevee avoiding Dratini before it can use Wrap, with Eevee using Tackle when the chance presents itself. Blue is quite stubborn and generally refuses to order Dratini to do anything other than it's Wrap attack, but Eevee has an advantage in speed and gradually dealing the Dragon-Type some pretty significant damage. On the other hand, a single good Wrap attack could still turn things around in Blue's favor. It's in your and Eevee's best interest to keep cautious and wear Dratini down little by little.

Unfortunately, you fail to account for Eevee's speed beginning to decline. A battle of attrition seemed like your best bet, but despite the fact that your Pokemon hasn't taken any damage she's still wearing herself out keeping her speed up to avoid getting caught up in Dratini's Wrap. Blue's Dratini is finally able to get a good hold of Eevee after her movement slips up, and puts the squeeze on your Pokemon.