I can't give you any specific advice about Elmer's Court as I've never stayed there but you are in a fantastic part of the country, particularly in July. Lymington is a small, historical coastal village in the New Forest, one of the last great forest's in the UK. It is itself full of history and interesting sites,
https://en.wikipedia.org/wiki/New_Forest.
Will you have the use of a car? Nearby Bournemouth, Southampton, Winchester, Portsmouth and Salisbury are all excellent places to visit and can easily fill up your week. Then there's Stonehenge, Corfe Castle, the Jurassic Coast and even Bath is within reach. Public transport can be used to visit most of these places however a car would be more beneficial.
I can recommend two fantastic places to eat,
https://www.thepighotel.com/brockenhurst/ and
https://www.montaguarmshotel.co.uk/. They're not the cheapest of places but the food is excellent.
There is so much to see and do it is probably best to list your interests and means of transportation in order to get a more specific recommendation of sightseeing suggestions.