There are some Mohelim who do not come to check the baby at all. The truth is that it is usually possible to ascertain with a few questions over the phone whether everything is OK. However, there are frequently surprises, and to play it safe, it is a good idea to come and check the baby beforehand. And that is why many Mohelim come check the baby before the Bris.
Some parents just look at this as an opportunity to meet the Mohel. While that is true, there are a few things the Mohel will want to check for:
Yellow - Jaundice
The Halacha is that if a baby is yellow, the Bris must be postponed until the yellow goes away. Medically, this yellow is called Jaundice. The Mohel will want to see if the baby has jaundice. When a baby has a high bilirubin level, he will have a yellowish tinge. This yellowness will depend on how high the bilirubin is. Many babies have increased bilirubin levels, but if the baby has a bilirubin level of over 13-14, the Bris will need to be postponed until the bilirubin goes down. (This varies somewhat depending on the Mohel).
Usually the Mohel can tell how high the bilirubin is based on the color alone, but on occasion they will refer you to someone who does bilirubin testing to get an exact result.
Many parents may notice that their baby looks very yellow in the face. This doesn't mean the baby is yellow. The face is not really where a Mohel judges from, rather the stomache, legs, and backside.
Practically, if your baby is too yellow, and you would like it to go down so that the Bris can be on time, the baby needs to drink from a bottle (only during the period where you are trying to get the yellow down). Some newborns have a hard time nursing, and they tire out from it, and have no strength to nurse anymore. At that point, even if they are given a bottle, they won't drink it because they are too tired. However, if you start the feeding with a bottle, it is easier for them, and they will eat. Eating is the best way to flush the yellow out of the system.
The Halacha is that if the baby has an eye infection, the Bris will have to be pushed off. The amount of time it will have to be pushed off is either until the baby heals, or until a full week after the baby heals, depending on the severity of the infection. (Some Mohelim will always wait a full week)
The Mohel will want to see if the baby has an eye infection. When a baby has an eye infection, the inner eyelid will usually be red, and/or the eyelid will be swollen, and the baby will have difficulty opening his eye. There will also usually be a strong discharge from the eye, generally yellow or green.
Frequently, at this age, the baby will develop blockage in his tear ducts, which will create a discharge. There is no problem with doing a Bris on a baby with a blocked tear duct, but the problem is that this can frequently develop into an eye infection. Practically, if discharge is the only symptom, the Mohel may tell you to give him eyedrops just to make sure that it doesn't develop into an eye infection.
This being said, it is important to let the Mohel know if you see any of the above symptoms, as early as possible.
Strength and Reflex
The Mohel will want to see if the baby has adequate strength and reflexes for a baby his age. The way this is checked is by seeing if the baby can hold up his head at all, if he has strength in is arms to be pulled up, how strong his legs are, and if the baby will grab the fingers of the Mohel. Generally speaking, a healthy baby should be able to do these things. If not, there is concern that the baby might be ill, and the Mohel may send you to your doctor for further consultation.
Type of Bris
Although most babies are born with regular bodies, some are born with certain conditions which need a special approach for a Bris. Without getting involved in the complications, some of these conditions are hypospadias, chordee, hydrocele, etc. These are not something that the parent would necessarily notice, and sometimes they will not even tell the parents in the hospital. So the Mohel will want to check to see if anything is unusual.
Aside from certain complications, every baby has a different anatomy, and while many Mohelim have a one-size-fits-all approach to a Bris, there are some who will take a personalized and individual approach to each Bris, depending on the specific anatomy of that baby. Therefore, such Mohelim will want to see the baby's specific anatomy so they can plan out exactly how they would like to perform this specific Bris.
The Mohel should give you instructions of what to bring to the Bris, and a list of Kibbudim for the Bris. You can read more about this in my section: What Do I Need To Know? - The Week Before the Bris.