I know how tempting that is! I was SO tempted to get my daughters ears pierced when she was a baby (especially since everyone was always mistaking her for a boy!--Even if she was dressed in pink it would happen!)
There are a few problems with getting any small child's ears pierced.
First of all, it is a form of body modification that's not being left up to the "owner" of that body. Second, I have several friends who got their ears pierced as babys/toddlers & now have VERY STRANGE looking holes because their ears, of course, grew & the holes are now either WAY too high on their lobe or WAY too low. One of my friends has holes that are SO high that she can't wear most small hoops & thinks that it just looks so silly that she has refused to wear earrings in years.
Another problem is that when anybody gets their ears pierced w/ a "gun" that it uses blunt force trauma to shove the earring through the skin. It's very painful & the stud traps a lot of bacteria between itself & the ear, which gets shoved further & further into the hole when the earring is "twisted" several times a day.
IF you choose to have it done, I HIGHLY suggest going to a good tattoo/piercing studio to have it done. The equipment is 100% sterilized (not so w/ a gun--there is NO way to properly sterilize plastic & piercing guns have been associated w/ Hep. B outbreaks!!!). It's done w/ a very sharp needle & a pair of clean hoops, which trap much less bacteria than studs do.--Many places will even let you bring your own if you so choose.
We left this choice up to our daughter, who decided to get it done back in May (she was 5). I took her to a tattoo shop & she did awesome! She was SOOO proud of herself! It was a huge accomplishment for her b/c it was something that she was very nervous about but really wanted done. Her ears never got even the tiniest bit red, inflamed, infected....nothing! What a huge difference!--I have my ears pierced quite a few times & they were all done w/ a gun & I recall very well how they throbbed for days.
Anyway! It is up to you as her mother to decide what you want to do. But if you do choose to have it done, again, I really hope that you can find a tattoo shop to have it done in. You'll pay a little more, but it's well worth it!
Good luck!!!