What's a good way to prevent students from saying I lost their paper?

First, get organized.

You need a good defence. Make sure that you are not at fault.

What I've done is to put student assignments in a folder and keep it entirely separate from any other papers I have. When I bring their assignments home, I transport them in a plastic bag to further distinguish them and prevent papers from falling out somehow. For situations where the class turns in lots of assignments and students don't always pick up their graded assignments, papers can get out of order, so I make sure to keep each assignment in a separate folder, or I put all graded assignments in a single stack alphabetized by last name. I know with full certainty that students' assignments are in my "to-grade" folder, or my "to-return" folder, and nowhere else.

Second, respond to the student

If they say you've lost their paper, give them the benefit of the doubt by looking through your stack. If it's not there, you know it won't be anywhere else, short of a devious third-party intervention, and you can tell them so. :-)

Alternatives

• Use an electronic submission dropbox (e.g. Blackboard, D2L, etc.).
  • You could have students email their assignments to you, but there's always the possibility that it gets caught in your spam filter. If you did this, you should send them an email ASAP to confirm that you received their submission.
• Keep all papers in a single room: never take them home to grade, etc.
• Keep a list of students who submitted their work as they turn it in.
• Scan in assignments, keep an electronic copy and send the copy to students (idea from @MadMyche, in the comments below).

Have the students write date, their name and the title of the assignment into the lower right corner of the first page.

When you receive a paper, put a stamp/short signature next to that, cut off the corner, and hand it back to the student. It'll serve as a receipt. There are even stamps with an "auto-increment" feature, which produce serial numbers.

Now, no one can claim you've lost some paper which has never been in existence, and no one can use the cut-off corners for a different assignment (you don't even have to collect them when returning the graded papers).

Next step: do not lose papers.

There are many good answers above. However, for your particular situation (only physical papers, and being realistic) realities of life is that submission within the classroom is rather hectic and you can't check/verify every student submission on the spot.

The way I suggest is simple: you just need to ensure that nothing is/was lost by you. In order to do so:

I place a large envelope (or even a box) on the table, visible enough to ensure no one can take something from it. and require students to place their work in the box/container. All what I need to do is to count the submissions and write down the count before I dismiss the class. If you want even more reassurances, tell the students the count (the latter will have the extra benefit of highlighting how may submissions are missing with out pointing out anyone or spending too much time verifying).

After grading, I know how many I had and how many I graded. If the numbers match, I haven't lost anything.

So, that solution should take you about 60 seconds to implement and no one can possibly claim that their work was lost as long as your numbers match.