Large Language Models (LLMs) face significant computational and memory constraints when processing long contexts, despite growing demand for applications requiring reasoning over extensive documents, multi-session dialogues, and book length texts. While recent advances have extended context windows to 100K-1M tokens, such approaches incur prohibitive costs for resource constrained deployments. We propose BudgetMem, a novel memory augmented architecture that learns what to remember rather than remembering everything. Our system combines selective memory policies with feature based salience scoring (entity density, TF-IDF, discourse markers, position bias) to decide which information merits storage under strict budget constraints. Unlike existing retrieval augmented generation (RAG) systems that store all chunks, BudgetMem employs learned gating mechanisms coupled with BM25 sparse retrieval for efficient information access. Through comprehensive experiments on 700 question answer pairs across short (237 tokens) and long (5K-10K tokens) documents with Llama-3.2-3B-Instruct, we demonstrate that BudgetMem achieves remarkable results on long documents: only 1.0% F1 score degradation while saving 72.4% memory compared to baseline RAG. We validate our approach through budget sensitivity analysis (testing 7 budget ratios), naive baseline comparisons, and document length analysis, showing that BudgetMem's benefits increase with document length. Our work provides a practical pathway for deploying capable long context systems on modest hardware, democratizing access to advanced language understanding capabilities.