Recent advances in large language models have revolutionized many sectors, including the database industry. One common challenge when dealing with large volumes of tabular data is the pervasive use of abbreviated column names, which can negatively impact performance on various data search, access, and understanding tasks. To address this issue, we introduce a new task, called NameGuess, to expand column names (used in database schema) as a natural language generation problem. We create a training dataset of 384K abbreviated-expanded column pairs using a new data fabrication method and a human-annotated evaluation benchmark that includes 9.2K examples from real-world tables. To tackle the complexities associated with polysemy and ambiguity in NameGuess, we enhance auto-regressive language models by conditioning on table content and column header names – yielding a fine-tuned model (with 2.7B parameters) that matches human performance. Furthermore, we conduct a comprehensive analysis (on multiple LLMs) to validate the effectiveness of table content in NameGuess and identify promising future opportunities. Code has been made available at https://github.com/amazon-science/nameguess.