This is still an issue in Jenkins 1.450

I was able to reproduce using a vanilla OOTB Jenkins install.

Steps to repro:
1- Install Jenkins
2- Go to Manage / Configure page, click Advanced... button at top of screen, set Build Record Root Directory to /scratch/jbuildroot/${ITEM_FULLNAME}/builds or something like that, click Save
3- Create a test job with a shell script section, do a few runs, note the logs are saved to /scratch/jbuildroot/JOBNAME/builds
4- Rename the test job. Note there is a new dir at /scratch/jbuildroot/NEWJOBNAME/builds but it has one empty run. The old runs are still under /scratch/jbuildroot/JOBNAME/builds
5- Restart Jenkins via yourhost:8080/restart
6- Note the test job runs are gone because the Build Records were not moved to the new dir.

One shell script workaround for renaming the fileystem parts of job (presumes a path like ${ITEM_FULLNAME}/builds is listed below. Save as 'renamejob.sh'.

First, backup your original job name directory, just in case.
Next, rename the job in the Jenkins UI.
Then run this script
Last, restart Jenkins (required).

Usage: renameJob.sh pathToBuildArtifactDir oldJobName newJobName

#!/bin/sh -eu

JOBROOT=$1
OLDJOB=$2
NEWJOB=$3

if [ ! -d $JOBROOT ]; then
echo "ERROR: directory at JOBROOT=$JOBROOT does not exist"
exit 1
fi

if [ ! -d $JOBROOT/$NEWJOB ]; then
  mkdir -p $JOBROOT/$NEWJOB
#echo "ERROR: newjob directory at $JOBROOT/$NEWJOB does not exist"
#exit 1
fi

if [ ! -d $JOBROOT/$OLDJOB ]; then
echo "ERROR: oldjob directory at $JOBROOT/$OLDJOB does not exist"
exit 1
fi

echo "RENAMING JOB from $OLDJOB ==> $NEWJOB"
set +e
rm -rf $JOBROOT/$NEWJOB/builds/*
mv $JOBROOT/$OLDJOB/* $JOBROOT/$NEWJOB
set -e

echo "RENAMING OLD JOB TO END IN .old"
set +e
mv $JOBROOT/$OLDJOB $JOBROOT/${OLDJOB}.old
set -e

echo "UPDATING RUNS..."
for RUNDIR in `ls -d $JOBROOT/$NEWJOB/builds/2012*`; do
echo "... UPDATING RUN AT $RUNDIR"
if [ -f $JOBROOT/$RUNDIR/junitResult.xml ]; then
sed -i "s/$OLDJOB/$NEWJOB/g" $JOBROOT/$RUNDIR/junitResult.xml
fi
if [ -f $JOBROOT/$RUNDIR/build.xml ]; then
sed -i "s/$OLDJOB/$NEWJOB/g" $JOBROOT/$RUNDIR/build.xml
fi
done

echo "REMOVING .old JOB (this may fail if more work is needed)"
rmdir $JOBROOT/${OLDJOB}.old

This is still an issue with Jenkins v1.454.

I noticed that in Jenkins v1.454, renaming a job does not even create the build record root dir entry. For example, if I have build record root dir set to /scratch/jbuildroot/${ITEM_FULLNAME}/builds and I rename a job 'foo' to 'bar', then /scratch/jbuildroot/foo exists, but the rename does not create /scratch/jbuildroot/bar.

I think in a previous version, this directory was at least created (though it was empty), but in v1.454, I dont see it created anymore.

Updating the summary to reflect the new information.

Duplicates JENKINS-17138 (it was resolved using that issue ID)