To begin with the cost saving project for HR department, as a first stage you need to know the current or historical spending areas by HR department.
One of the area being Recruitment. So, just to take recruitment factor answer the below questions.
Stage - 1: Data Collection
Total Recruitment last year or couple of years
Cost for recruitment (cost paid to agencies, referral bonus)
Sources for recruitment (consultancies, referal, campus hire, advertising)
Also, termination within 3 months, reasons for it.
Collect as much as possible details on recruitment
Stage 2: Now, look for opportunities
If you pay more to consultancies, check if those positions can be filled though employee referrals.
Check whether you have clear process on recruitment (effective interview techniques, effective screening techniques and the skills to identify the sustainability of employees)
Stage 3: Implement and evaluate
Once you find the areas of improvement, please implement.
Rather than saving cost to company directly, HR can save a lot by designing clear HR polices and procedures.
Do you have policy leave encashment for employees? Any limit on that?
How about reimbursements and Expenses claim policies?
Are they cost effective now? Please understand your HR department and functions completely and the cost associated to it.
Sometimes, more than cost saving HR can play vital role in improving quality and efficiency.
Good luck for your project.