I'm often asked what age is appropriate for a child to start guitar lessons. I completely understand where this question comes from, as it usually stems from parents who have heard other teachers recommend they wait until the child is older. There are reasons for this, and I feel I should address these first, as they are valid concerns on the part of the teacher.
To begin with, young children - especially children between 5 and 10 years old - have much smaller hands and fingers than their adult parents. Naturally this poses a problem if the instrument is a full-size guitar. There are half-size and 3/4-size guitars available from reputable makers that are scaled to fit a child's hands and the reach of their smaller arms. These guitars will feel much more comfortable to the child and they will find they are better able to 'handle' the instrument.
However, there is more to the concern than just the instrument's size. Young children are still developing their fine and gross motor skills, both of which are required to play the guitar - especially fine motor skills. If the child is not able to control their fingers and press firmly on the strings, they will not be able to produce good tone. Students need to keep their fingers arched and relaxed for relatively long periods of time (up to a few minutes, which is taxing on small hands that are not yet fully developed). One common workaround for this is to have the child play with nylon strings, which are generally slightly thicker than steel strings and far less painful for soft, young fingers.
Young children may also have difficulty reaching the frets on the guitar. Playing a note in the first fret and the third fret simultaneously can be a real stretch for them. One way to bypass this problem is by teaching them to play higher on the neck. This can be done while still teaching notes and scales and chords. Some teachers may opt to have the student use a capo, which raises the pitch of all the strings and allows the student to play higher on the neck of the guitar while still reading music intended to be played lower on the instrument.
Another concern is the attention span of young students. A 7-year-old should not be expected to be able to focus on much more than a short passage or two at a time. Some students are more capable of focusing their effort on playing than others, and the amount of music you give them to practice will likely vary between students. This is one of the most stressful factors for teachers of young musicians, and I feel is also the least important. Lessons are often structured in 30-minute sessions, which is fairly standard for private lessons and even many (I daresay most) college-level lessons. Understanding that your young student may not be able to apply focus for that amount of time is key to maximizing the effectiveness of your lesson. If a student becomes tired or disinterested during a lesson, it is time to change the focus. Perhaps move on to another technique or song, or integrate other concepts into the lesson. Puzzles that teach young musicians to recognize intervals might be a valuable use of that time. At the end of the lesson, you might go back and play through exercises the student has already completed, thus cementing those skills into their minds and giving them a sense of mastery over their musicianship.
Finally, a child's ability to practice regularly each week is a major issue. If the child is unwilling or unable to practice a reasonable amount of time, they will progress much slower than when they do practice, if they progress at all. If your child shows interest and demonstrates the ability to handle the responsibility that practicing requires, then they may do well with guitar lessons.
Having considered these issues, I've found that the only way to really know if a child is ready for guitar lessons is to put a guitar in their hands and see what happens. I know many well-meaning parents run to the nearest "Buy-Mart" and purchase a colorful and properly-sized, but poorly manufactured and barely tunable instrument. I do not recommend these guitars for anyone intending to use them for lessons. However, I do see the value in allowing a child to explore this 'toy guitar' to see how deep their interest really is before investing in a proper instrument. This way, parents can gauge their child's interest without spending much money initially.
I teach young children to play guitar and make accommodations with almost every student, including the methods described above. Each child will have different needs and abilities, though, so your teacher must take all of this into consideration. In some instances, a teacher may have to decline to teach a child. Making that call will differ between teachers, and if a teacher tells you they won't teach under a certain age, don't be afraid to ask why. They may even have other ideas for you that may be helpful.