Is it somehow possible to resize a partition over ssh? Or is it absolutely necessary to umount it first?

Which partition? Can you umount it over ssh and then resize it? E.G. If you log in as root, and init to single user mode, you can umount the /home partition. Root doesn't need the /home partition and in single user mode, no other use is logged in who does.

This is something that is probably safer to do locally.

sorry, I meant root(/) partition. Or is it possible to install LVM over ssh and then resize and make new partitions over ssh?

Anything is possible.
Would you do it to a system you care about ??? ... that may be entirely different.
Most filesystems probably allow online enlarging these days. Personally I don't do it. Reducing the size online - hmmm ...
LVM is irrelevant to this discussion - the filesystem still has to support resizing.

I suspect single user won't work over ssh - last I looked at a Ubuntu system, it didn't load the network in runlevel 1.

i think, this step should be done local. it is to dangerous to do it remote.

it is my parents desktop computer(Ubuntu) in another town. They have a netbook as well so it is nothing catastrophic if the desktop computer is out of order for a while. This Ubuntu desktop computer has three partitions: 1GB swap, 49GB /home and 150GB root(/) partition. The problem is that this /home partition has become full(mostly backups of movies, nothing critical). On the other hand, the root partition has 50GB of free space. I would like to copy the content of /home partition into a / partition. Then delete /home partition and finally extend / partition. Is it possible to install LVM now or is my only possibility to try partitioning over ssh?