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I think this is a nontrivial question, because to avoid embarrassment or singling-out, I cannot ask students about:
Ideally I would like some nice questions with numerical answers where we will get non-embarrassing outliers, so one student can verify that yes, he is the reason the mean of this data set is so crazy -- without feeling awkward or ashamed or really having any emotional attachment to it whatsoever.
If you already have the class roster, you can look at the students' names and see if you can find any outliers in measures associated with their names. Numbers of letters, longest run of consonants, total number of consonants, largest difference between letters in the first and last names, etc. It would depend on your data set.
Number of items they are carrying on their person (not counting items of clothing, but counting anything in pockets or purses, but not book bags). There may be outliers, especially within subgroups. And differences in the means of subgroups.
Get to the classroom early and note the time each student arrives for class. You can do this manually, or with a digital camera that will timestamp pictures. As people may not want their pictures taken, you could assign an index to each student or group of students as they enter (a number on a slip of paper). All you need do is photograph the slip of paper. They can hold onto the paper, so they can be associated with the time. Or, write the index on the chalkboard and photograph that, and ask them to remember the index they are assigned. Or, completely disassociate the person from the index and just record the times to keep the discussion focused on the data rather than the people.
Where were they when they last got into a car? Find it on Google Maps, ask for directions from there to your campus, and record the distance Google Maps gives you.
The results of a game involving some trivial-to-explain but variably difficult task. For instance, the number of words they can think of in 30 seconds that begin with one letter and end with another. Have them choose the letters from a bag filled with Scrabble tiles. Or a simple script written in any scripting language (or even Excel script) that assigns the beginning and ending letter. Have them pair up so that one person just says as many words as he or she can think of and the other counts and is in charge of time. Each person has a different pair of letters, making it possible for some people to get a very difficult combination. Tweak the parameters here as desired.
Alternately, you could grab some data from the DASL Data and Story Library. :
You can search based on the method of analysis you're interested in working with, but this also relates to points of interest in the different sets. For instance, go to "stories by method" and then choose "Outlier" -- it will give you data stories with outliers.
Somewhat related: I stumbled upon this resource when I was preparing to present Tinkerplots as a technology for visually exploring sets of data. A quick example of this use is in this blog post and video, regarding some interesting attributes of the set of passengers of the Titanic. The question being explored is "Who survived the TItanic?"
EDIT 4/8/15
I began teaching Statistics with the same difficulty that you had. I started with basic sorts of questions (e.g., age, height) but my students weren't really responding to these types of questions. I sought to find a way to have them feel more connected to the data.
I began with questions about text messaging. (1) How many different people did you text on Sunday? How many total texts did you send on Sunday? These actually led to other interesting questions that we had to have as a class. Do we include group texts? Do we include picture texts? These types of questions that lead to the the exact discussions that you want to have with your students about data collection.
As this line of questioning developed, I started to have my students survey their own friends and contacts about their texting habits. This leads to other interesting data collection questions that we could discuss as a class. While we were able to get numbers we could work with, we had to be especially careful about what our population was, and how our data collection methods might not have been ideal. My last lab asked them about their flossing habits and total cavities. This didn't really provide the wide variety of data that I was hoping for, but it was suitable for my purposes.
The best thing about all of this what it opened up my students to start thinking about their own questions. They came up with many more questions than I could have. How many days a week do you work out? was a popular question. Shopping habits (online vs. in stores), coffee consumption (Starbucks or Dunkin), and QBR ratings for quarterbacks were all places that students pulled data from. I even had one student track her daily coffee consumption and sleeping habits for a month to use as a data source.
I'm not sure what level of student that you have, but for my college students of traditional age, once I was able to ask some questions about what they actually did in their lives (i.e., texting) it opened them up to the possibility of what other aspects of their lives Statistics could answer. I think if you can do this, then you'll help them to not only learn Statistics, but to teach them how they can use Statistics in their everyday lives.
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I utilized a lot about their phone habits when I taught statistics recently. Number of total texts sent in one day, number of unique recipients of texts you sent, number of Instagram messages, etc. I can provide more details in an edited answer when I'm back on my computer
How many hours a day do you read
