How is udhcpc executed and how to change it?

Your system seems a lightweight version/variation of Debian, based in busybox.

busybox is typically used either for recovery medium, or for embedded systems with limited resources.

For modifying the parameters, you can invoke udhcpc automatically.

You can change /etc/network/interfaces as:

iface eth0 inet manual
   pre-up /sbin/udhcpc -R -b -p /var/run/ -i eth0

As for running another DHCP client, you would have to install it; however you would have to switch it with udhcpc unless you have other interfaces.

Bear in mind as udhcpc is part of busybox, it is just a link to a global binary that provides you with a work environment, and as such, you won't save any space switching DHCP clients.