ORA-01555: snapshot too old: rollback segment number with name "" too small This error appears when I am trying to move LOB segment from one tablespace to another one.

Also during exporting the same error appears.

What can be a problem?
Per the Oracle Docs and MetaLink support note 66431.1, the standard undo mechanism is not used by LOB segments, which have their own page versioning (controlled via PCTVERSION and the LOB_retention_clause).

"Snapshot too old" generally means that a transactionally-consistent version of your data could not be hold long enough because of lack of space. It might be that your PCTVERSION setting is too strict and effectively causes any new update to invalidate the consistent version(s).