It turns out I was wrong. Two weeks ago, for the first time I did initiatories for my own relatives who I had found myself, and currently I have ordinances reserved for 37 people on Family Search*. If you believed this myth too I can almost guarantee that you can find family names if you're willing to work for it, and I'm going to tell you how.
Step 1: Start with yourself and work back.
I know you probably want to get on Family Search and start where the family lines end, but you don't want to do that. If you haven't already, make an account on Family Search (your login is the same as your lds.org login). Make sure you are on there and that your dates and relationships are correct. Take some time to write at least a brief life history of yourself and upload it as a story to the "Memories" section, or add some photos. Then move back to your parents and check their pages for accuracy, and add some memories or pictures about them too. Then move on to your grandparents and check their pages for accuracy. Keep going.
Ask family members if they have old photos or life histories and add these to the Family Search Memories as well. These things are treasures and it's a real treat to be able to look at an ancestor on Family Search and see what they looked like or read about their lives. Doing this will also help you get comfortable using Family Search.
As you're getting started doing family history, it's really helpful to do indexing. Go to FamilySearch.org and click Indexing to get started. There are a few reasons to do this: One is that it helps people who are doing their family history (including you!). If a set of records is indexed, it is a lot easier to find people because you can just type their name to search for them instead of reading through the records page by page. Another is that it will help you get a sense of what information is helpful while doing family history and it will help you learn about the types of records and information that are out there. Plus, it's actually pretty fun, and much more fulfilling than Facebooking.
Step 2: Add sources!
As you do this, make sure to document whatever information you have with sources, especially as you move back generations. Find birth certificates, death certificates, and census records. A lot of these are available on FamilySearch.org or their partner sites (you can get accounts with MyHeritage, Ancestry.com, and Find My Past for free through Family Search, which gives you access to many great records!). The reason for this is so that anybody else who gets on Family Search (including you) can see why the dates, places, and other information you have added are correct. A lot of what I did for the first few years of doing genealogy was adding sources to people who were already on Family Search and had their temple work done, but there was nothing to verify any of the dates and places.
Don't assume that just getting a date is good enough and you can stop there. The more information you find, the better. A couple of months ago I found the youngest child of a family in a census record. I had already found a lot of information about that family, and if I had stopped before finding that record I would have missed him.
As you move back in time, you will probably find that people's pages get less accurate. I think for me this happened a lot more around six generations back, but it could be earlier for you. By the way, by that point I wasn't systematically going through each generation, but instead I was focusing on specific lines as led by the spirit. Sometimes Family Search will have a red exclamation mark to point out blatant problems like children born after their mother died, marriages at extremely young ages, children born before the parents were old enough to have children, etc. But other problems you will have to find yourself. Here is one example in my family tree:
One obvious problem is the three sets of parents. The marriage in Norway is probably not accurate because Norway to Denmark is a long distance to travel at that point in time, and the other two sets of parents might be a duplicate of each other. Possibly none of the parents are correct. There isn't much information there. At some point I will need to find sources to figure out where and when she was born and who her parents were, but right now I'm still working on her descendants.
I have found that as I correct these problems I tend to find more family members who need temple work done. I would recommend starting with your American or British lines because the sources will be in English and a lot of them are indexed, which makes things easier. Family Search often gives record hints to help you (although you should check to verify that they make sense before adding them as sources to your ancestor).
If you need help learning more about checking for accuracy and finding appropriate sources, go to your local family history center or talk to your ward family history consultant and ask for help.
Step 3: Do descendancy research.
You can also do temple work for descendants of your ancestors who were born over 110 years ago. So as you work your way back on a family line, don't just work on your own ancestors--look for any siblings and do their temple work, and also find spouses, children, and grandchildren for these siblings. It's actually a good idea to work on entire families at once because that way you don't miss people and you can find information about people through records about their siblings or children. For example, birth certificates or christening records often tell who the parents were, and many church events have godparents or witnesses who are also relatives.
Especially for people with a lot of pioneer ancestors, this is where you will find a lot of people who need temple work. A very useful tool is Puzzilla.org, a website affiliated with FamilySearch.org that generates descendancy trees of people on Family Search. It's only as accurate as Family Search, which has a lot of mistakes, but it can be helpful to use it to see good places to start doing descendancy research.
Here is my family tree on Puzzilla back 12 generations. Not all the lines go back that far, but many of them go back even farther.
I chose a random person from the twelfth generation and pulled up her descendancy chart on Puzzilla. This image only shows five generations of her descendants, but I stopped it at five because it would have taken a really long time to load all of her descendants. Already the people in this image number in the thousands. The yellow line shows how I am related to her. The red squares represent people who don't have any children entered into Family Search--children who could potentially be found and receive the blessings of the temple. Of course I'm not going to jump in and start looking at these people yet because I haven't verified the family lines back to Hannah Barsham yet, so I'm not actually sure if I'm even related to her. But look at how much potential for family history work there is here!
Step 4: Check for duplicates and merge them.
The information already on Family Search comes from a lot of different sources. Remember 20 years ago when your ward family history consultant was hounding you to submit your four generation chart to the church? That's on Family Search, and so are the ones your siblings, your parents, and your grandparents submitted, which results in a lot of duplicates. The trees from the IGI, PRF, and Ancestral File are also on there.
Extraction projects have been frequently used in the past to supply names to the temples since not everybody has their own family names, and this means that some people have duplicate records. For example, the brother of one of my ancestors had about seven different profiles on Family Search because extraction workers found his children's christening records and created new profiles for the parents with each child. I had to merge all the duplicates so that they appeared on Family Search as what they were--a single family.
To check for duplicates, go to the person's page on Family Search and scroll down to "Possible Duplicates" on the right side of the page. Click that link. Sometimes Family Search will show you some possible duplicates. However, sometimes for some reason it misses duplicates, so you should also do a search with the person's name and birth date under "Find" to make sure there are not other duplicates.
I have found that if there are duplicates, the people I find may have already had their temple work done, and there's not really a point in doing their temple work a second (or third, or seventh) time. But I have felt often that even if there isn't temple work to be done, these people appreciate having their records and families set in order on Family Search. I really believe that someday when Christ comes, Family Search is the "book" that we will be presenting to him with all the genealogy and temple work we have done, so it is important to get these families in order as much as we can.
Step 6: Find names and take them to the temple.
If you follow these steps, you will find family members who need temple work. (I don't like to refer to them as "names" because these are real people, but sometimes that's the phraseology that is used because it's convenient.)
Notice that one of the keys to all this is verifying the accuracy of what is already on Family Search and backing up everything you add with sources.
There are apps that will systematically look through your tree on Family Search to find people who need temple work done (for example, one is available as a Google Chrome extension at hopechestsoftware.com). I don't often recommend them simply because I think it's important to recognize that family names don't just appear out of thin air; they are found through research. However, there are people like me who have time and ability to do the research but can't go to the temple often, and there are other people who can go to the temple often but don't have so much time or ability to do the research--and these apps make it possible for both types of people to help each other. So if you want to use the apps, that's fine. I would, however, recommend that you take some time to look at the people who the app finds and see if there are possible errors or duplicates.
I'm not going to lie, it did take a lot of work to verify enough generations to finally find opportunities for temple work, and even now finding people takes time and effort. I've spent hours and hours reading through Danish parish records trying to find all the children in a family. But it was also fun and interesting because I learned so much about my ancestors, and now I am finally seeing the fruits of my labors. Already I have enough names to last me for a long time and I am planning to share them with my family members and ward, and I can see now that there is lots of potential. I seriously doubt I will run out of family history work to do in my lifetime.
*Update 12/10/17: Currently I have ordinances reserved for 94 people. I'm at the point where I'm considering sharing with the temple system or un-reserving some of the earliest ones so other users can reserve them because there's no way my husband and I can do all of this by ourselves.