You've helped your child with their first steps, first book and occasionally first aid. Now you can help them pick their first WoW server. Tell them to treat it as their first quest. Finding a server should be done in concert with your child picking their first character. If they are joining with friends who are also just starting out, they all need to choose the same server and faction (Alliance or Horde) in order to play together. If their friends have already started, the choice of server and faction may have been made. If your child's friends aren't too far along and another server looks more promising, encourage them to all join that one. For a group of friends playing the game together, it's best if they all start out together at the same level and stay at nearly the same level as much as possible.
The very first thing that needs to be decided is if your child wants to play on a Player versus Player (pvp or PvP) server or a Player versus Environment (pve or PvE) server. These two basic types of servers were discussed in the section, More Than Just Questing, but a synopsis is that on a PvE server, players from the other faction can only be engaged in certain areas or only if both factions specifically flag themselves for battle. The only things that can attack them at any time are computer controlled characters in the environment - hence the name. On PvP servers, any player can attack the other faction's players just about anywhere in the world (with some restrictions in neutral cities and areas). Even so, I would encourage most players to start out on a PvE server until they are certain they plan to play for a while and have a good feel for how to play the game. The very lowest level areas where new characters start are fairly well-guarded against the opposing faction even on PvP servers, but those only cover the first 10-12 levels. There will be characters 60-70 levels higher that have fun just "skinning newbies" for hours at a time. It's not against Blizzard policy on PvP servers. My recommendation is that your child's very first character should be made on a PvE server. Once they gain experience, they can start a new character on a PvP server to see if they like it.
Additionally, your child needs to decide if they want to play on a normal or "roleplaying" server. On a roleplaying server, they are generally expected to stay in character at all times in anything they type in-game. Also, the name of their character is also expected to reflect a roleplaying one. If I am to give some advice here, which can be happily ignored, it would be that unless your child is already well-versed in the World of Warcraft lore through the other Blizzard games and books, they should probably stick to a normal server for their first character.
There are some tricks to picking a good server. As mentioned in the Economics of WoW section, each server or "realm" as it is called in WoW is housed in one of Blizzard's (or Blizzard's partner's) data centers scattered around the world. Knowing this, one can use the wowwiki.com Realms List (unofficial list of existing realms and the data centers they are housed in) to pick a geographically close server. For US players, the US Realm Server List by Datacenter is the list to use. If you live the state of New York, for example, a Boston-based server (or maybe even a Dallas-based server) makes the most sense. The responsiveness of the game is usually good even for those players geographically far from the actual server, but the closer the server is to the player, the better that response is and the more consistent it is. On some days and nights, Internet traffic is such that those closest to the datacenters have a noticeably better playing experience than those who aren't. It's not the end of the world if a player located in Maine chooses a Seattle-based server, but there will be days when their play becomes sluggish. It may be momentary or last for several hours. You will probably also want to pick a server with the same time zone as the player, but it's not mandatory.
With a list of potential servers in hand, the next investigation is into the heath and population of those servers. This is where the World of Warcraft Census web site is very handy as an aid. (The numbers in the census are only representative of the whole and not absolute, guaranteed numbers. The census is taken by users that have a census taking add-on running in their game and periodically upload the census data to the web site.) First, look at the Quick Server Stats for either the US or European realms. (If you're handy with spreadsheets, the data from the table can be copied into a text (.txt) file where it becomes tab-separated. That file can then be imported into Excel. Just make sure to use Text rather than General cell format for the Online column.) Look at the reported population of the servers you are considering. I found that a good population range is from 12,000 to 22,000 players - give or take a thousand or so. Those with more players are overcrowded and tend to have mostly high-end server players. The cities are jammed with players causing the game to slow down in those areas and the dearth of lower-level players makes it hard to get groups for questing at those levels. On the other end of the scale, low-population servers may not have enough players to find groups for instances at any level. The economies of lower population servers can also be quite odd or severely broken.
Another aspect to look at is server age (looking at the "Online" column) versus population. That column lists the month and year that the server came online. New servers that have a large populations are typically "free transfer" servers. That is, Blizzard offered free transfers to players from one or more overcrowded servers to that new server. Additionally, that server was likely the suggested server for any new player starting WoW for the first time. These servers can be odd in that while the population is in the right range in total population, it's made up of two discrete groups: those under level 30 who took Blizzard's suggestion to use that server for their first character and level 80 players whose whole guild(s) transferred to the new server. The economies on those servers tend to be quite broken for a while but balance out as the lower level players start to hit the middle and higher levels. It can be hard to get started on these servers as well. One last item is that if your child is going to try a PvP server, it's good to pick a server where the population is fairly balanced or favors the faction that your child has decided to play.
Hopefully by now, the list of potential servers is fairly short. With that list, go to the Census Data page and enter each of the servers into the server pull down and press the Go button. This displays a list of players by class and race for both factions. Ideally, whatever class your child chooses, it is best if there isn't a glut of that class on that server. If they've decided to be a mage, for example, they will find life easier on a server where there is a lower percentage of mages than that found on other servers. The list can also be filtered by faction so listing the Alliance mages versus all mages is the number to look at (if your child has chosen to start as an Alliance member). Do this for the other servers on the list. If the percentage is lower on one particular server then that might be the one to choose.
With the final list of (hopefully just a few) servers, a good thing to do is to check each realm's forum on worldofwarcraft.com web site to get a general feel for how the other players on that realm feel about it. There are always moaners and complainers on any server, but in general, it is good to see if the realm seems to be relatively drama-free. If you see a number of guilds recruiting, some congratulations to a guild's latest achievement, and some good general banter, that is indicative of a healthy server.
The very first time a player logs onto WoW they are assigned a suggested server. Typically, these are low-population servers that are new as mentioned above. That realm is only a suggestion and there is a change realm button that lets a user pick the realm they wish. Check the short list of realms to see which of those, if any, are medium to high population at the "busy" hours of the evening - generally about 8 pm to 12 am EST (5 pm to 9 pm PST). High population servers can reach their maximum number of players and begin queuing players over that number that attempt to log in. Waiting in the queue for other players to log off is never fun. If the list of potential servers can't be whittled down to one, your child can start "throw away" characters on each remaining realm to see how it plays. They can also hike their character to the nearest large city to get a feel for the chat in the general text chat and visit the auction house to peruse the list of items and prices. (Staring a dwarf and running to Ironforge on the Alliance side or an orc and running to Ogrimmar on the Horde faction tends to be the most direct.)
If you're trying to stall for a little more time, tell your child that you won't set up the account until they have picked a faction (Alliance or Horde), a class (mage, rogue, hunter, etc.), their class specialty (or two), the primary professions they think they'll pursue and a short list of names they wish to call their character. They can use the WoW Armory to check if that name is already taken on the server on which they'd like to play. Don't be surprised to hear your child complain that all the good names are taken. Blizzard has a strict naming policy against "offensive" names that your child should be aware of. Getting the answers to those questions could easily take another day or two (or at least long enough to get the game installed).
Your child really should be able to give you at least tentative answers to the above questions. Their choice of faction (alliance or horde) determines some quests they will be offered as does their choice of class. As for specialty, that's really something that can be chosen later for most classes, so don't be too insistent they have that specified. For professions, a player can choose to play any of the primary professions, but certain professions make more sense for certain classes. A mage, for example, may want to pick up tailoring so that they can make magical cloth items that help them as a mage. A warrior might want to take up mining and armor crafting.
Once they've decided all that, they should be ready to make their character on the server of their choice. Congratulations! You child has spawned - in a manner of speaking.