If I own a property and use that property as office space for my small business. Since I'm losing possible rent money by using the property myself, should the rent value be considered a running cost for me from an accounting point of view?

Is there a specific name for this type of cost? Is it even considered a cost?

In economic terms it's called an "opportunity cost" - meaning that you have lost profit because you are not renting the space out to a paying client. It's good to keep opportunity costs in mind if you are evaluating alternative strategies for your business - such as moving your offices to another location. But for accounting purposes it's not a real cost - as you aren't writing yourself a check.

Just a nit pick -

From a cost/managerial accounting point of view, it's still considered an opportunity cost, and yes you would still take it into consideration. For that branch of accounting, that's exactly the kind of thing we would do - the "real" cost of something includes what you may have lost and it's definitely included in certain types of calculations.

From a financial accounting point of view, no it's not an expense. It's not going to show up on an income statement.

We do recognize such costs in accounting, absolutely. We also distinguish the difference between a cost and an expense. It's not an expense.