Gretchen and Boy are pen pals, and both of them will send stuff to each other, as shown in Pen Pals. For example, she sends him a letter, a bracelet that says "Boy is Awesome", and cookies, and he tries to bake cookies to send back to her. They also write letters exchanging their thoughts on their mutual hobbies. Boy protests when Girl calls Gretchen his girlfriend, but while he says it's gross, he seems sad that Gretchen has a boyfriend.
Gretchen's personality has yet to be seen in person but she seems to be a nice person due to sending to Boy a bracelet that says "Boy is Awesome", and cookies with a letter since they are pen pals.
Mouse and Cheese pretended to be Gretchen inside a metal suit to convince Boy that they are her to get more cookies for themselves and for the rest but it backfired in the end.