The amount of coins in the wallet is too small for sweeping I want to make a present to my friend for his wedding - a bitcoin paper wallet.

I managed to print it and send 100uBTC (0.0001BTC) to the paper wallet for a test.

Then I chose "Sweep paper wallet" and I got a message that the coins are there, but "the amount of coins in the wallet is too small for sweeping".

What is the minimum amount that I can sweep?
Sweeping is just sending all bitcoins to another address. As such, there is a network fee involved.

Network fees will come out of the value of the paper wallet.

Depending on what you are using for sweeping, you can vary the fees. Though 
a lower fee will take longer to complete:

https://bitcoinfees.21.co/

At the risk of dating this answer, I would currently allow 0.001 BTC in your friend's gift for fees for fast transactions, about 5 USD, (yes fees are pretty high right now.)