People choose particular names for objects, such as dog or puppy for a given dog. Object naming has been studied in Psycholinguistics, but has received relatively little attention in Computational Linguistics. We review resources from Language and Vision that could be used to study object naming on a large scale, discuss their shortcomings, and create a new dataset that affords more opportunities for analysis and modeling. Our dataset, ManyNames, provides 36 name annotations for each of 25K objects in images selected from VisualGenome. We highlight the challenges involved and provide a preliminary analysis of the ManyNames data, showing that there is a high level of agreement in naming, on average. At the same time, the average number of name types associated with an object is much higher in our dataset than in existing corpora for Language and Vision, such that ManyNames provides a rich resource for studying phenomena like hierarchical variation (chihuahua vs. dog), which has been discussed at length in the theoretical literature, and other less well studied phenomena like cross-classification (cake vs. dessert).