cat /var/log/php5-fpm.log
[10-May-2017 16:17:22] NOTICE: configuration file /etc/php5/fpm/php-fpm.conf test is successful
[10-May-2017 16:17:22] NOTICE: fpm is running, pid 925
[10-May-2017 16:17:22] NOTICE: ready to handle connections
[10-May-2017 16:17:22] NOTICE: systemd monitor interval set to 10000ms
[10-May-2017 19:11:08] WARNING: [pool www] server reached pm.max_children setting (4), consider raising it
cat /var/log/lighttpd/error.log
2017-05-10 16:17:20: (log.c.164) server started
The cache size set to 32MB.
The OpCache has 618 scripts loaded just before the swapping begins. Do you need this huge list?
It is all I could copy for now - my system is locking itself up really fast. (I had to reboot twice in order to catch those files 
)
UPD: I’ve found this
Note
Lighttpd is not supported with Nextcloud, and some Nextcloud features may not work at all on Lighttpd.
here https://docs.nextcloud.com/server/9/admin_manual/issues/general_troubleshooting.html
Shall we move nextcloud to nginx? How can it be done on DietPi?
UPD2: I’ve switched to nginx web server using dietpi-software package installer. It was really easy to do and it works much more stable and much faster than lighttpd. At the moment I can successfully use netatalk (TimeMachine backup), nextcloud with ~70GB of files and RPI-Monitor. All of this with 170-200MB of RAM consumed (on RPi3 board). Let’s see it in the long run. So great! Thanks again for this great distro and support.
Thanks for your help!