Patall et al. (2010) has demonstrated that giving students choice in homework has the following advantages: (1) higher intrinsic motivation to do the homework, (2) feeling more competent regarding the homework, (3) better performance, and (4) better completion of the homework.
My experience is that these advantages indeed happen. After participants have been doing homework after training sessions we always come back to the homework during the next training session. For that purpose we use an exercise we call homework-walking which we usually do right after the lunch break. We ask participants the following: "Choose another participant and tell each other which of the homework suggestions you have chosen to do and why you have chosen them. Also tell each other how the homework was useful for you." Then we invite them to do this exercise while taking a 20 minute walk outside.
Doing the exercise like this has several advantages. The participants are socially, physically, and intellectually busy, all at once. Research has show that walking is quite healthy. A 20 minute walk can lead to much greater brain activity. Such a walk can become even more fulfilling and useful when you are also talking about an interesting topic with someone you like. On top of that the walk can become even more valuable when participants are walking together in a beautiful natural environment. Recent research has shown that being in such an environment has a vitalizing effect.