The deliberation process for incorporating an HR Case Management Software is a crucial determinant to an organization's operational efficiency, compliance profile, and overall employee experience. Strategic budgeting for such an investment demands beyond the mere monetary considerations, extending into a comprehensive evaluation of organizational needs, cost-benefit analysis, and potential return on investment.

Framed under the lens of Industrial Organization Economics, the decision to adopt an HR Case Management Software is parallel to a firm's strategic decision-making process when choosing to enter a new market or launch a new product. Akin to the Market Structure Analysis, it requires astute awareness of the organization's human resources landscape, the underlying challenges, and the potential software solutions that best address these issues.

The first step in this process is a thorough Needs Analysis. This process, analogous to the demand estimation in economics, involves identifying the specific HR case management issues the organization faces and aligning the requisite software features to address them. Considerations may include case tracking, knowledge management, self-service portals, analytics and reporting, etc. Understanding which features are mission-critical versus those considered 'nice-to-have' will help delineate the budgetary boundaries.

Subsequently, the Cost-Benefit Analysis becomes a pivotal financial tool. Here, opportunity cost - a core economic principle - becomes highly relevant. The cost of obtaining a software should be weighed against the benefits of its implementation. Benefits can be both tangible, such as improved operational efficiency and cost savings; and intangible, such as enhanced employee satisfaction and morale.

When analyzing the opportunity cost, organizations should consider the potential expenditure on dealing with HR cases manually, the time spent by HR professionals on these tasks, and the potential legal costs of non-compliance or mismanagement. The monetary equivalent of these variables might provide a clearer understanding of the budget required to alleviate these issues.

Return on Investment (ROI) is another key concept in this strategic budgeting process. Essentially, ROI is a performance measure used to evaluate the efficiency of an investment. For HR Case Management Software, ROI can be calculated by dividing the net benefits (benefits minus costs) by the costs of the investment. It's a simple, yet robust financial metric that can provide a snapshot of the potential profitability of this investment.

Moreover, the concept of risk and uncertainty, fundamental in decision theory and financial mathematics, comes into play when budgeting for a software solution. It's important to factor in potential risks such as implementation failure, software compatibility issues, employee resistance to new technology, or even hidden costs. Conducting a risk assessment and establishing a risk management plan can be instrumental in mitigating these uncertainties.

As an extension of this concept, the principle of diversification from portfolio theory can contribute to a risk-averse budgeting strategy. Applying this principle to HR Case Management Software would mean considering different vendor offerings to effectively spread the risk.

Negotiating contract terms is another strategic step. Here, drawing from contract law, organizations should look into payment structures and terms, maintenance and support provisions, and potential exit clauses. It's prudent to consider the Total Cost of Ownership (TCO) rather than just the upfront cost. TCO includes the initial purchase price and the ongoing costs of use, maintenance, and eventual retirement of the product.

Finally, the budgeting process is not a one-off affair but a cyclical process necessitating regular reviews and adjustments. This conjures up the image of a feedback loop in systems theory, where the outputs of a system are routed back as inputs. In our context, it means revisiting the budget and revising it based on ROI, user feedback, and evolving organizational needs.

Strategically budgeting for HR Case Management Software is a multi-faceted task, calling for insights from economics, law, mathematics, and systems theory. However, when done right, it sets the stage for the successful implementation of a tool that can drastically enhance HR operations, compliance, and the employee experience.