You are free to fill in setting details about Neverwinter or other towns or tribes. But please refrain from adding more connections to your destination town -- Phandelin. Doing that puts a bit of a burden on me to come up with details on the spot that can interfere with the intended plot.

The tavern thing is fine. Your character would be referring to Stonehill Inn, run by Toblin Stonehill and his wife and kids.

A bit on Phandelin (with a link to the OP):

The frontier town of Phandalin is built on the ruins of a much older settlement. Hundreds of years ago, the old Phandalin was a thriving human town whose people were firmly allied with the dwarves and gnomes of the Phandelver's Pact. However, the same ore horde that sacked the mines at Wave Echo Cave laid waste to the settlement, and Phandalin was abandoned for centuries.

In the last three or four years, hardy settlers from the cities of Neverwinter and Waterdeep have begun the hard work of reclaiming the ruins of Phandalin. A bustling frontier settlement has grown up on the site of the old town, and is home now to farmers, woodcutters, fur traders, and prospectors drawn by stories of gold and platinum in the foothills of the Sword Mountains.

Unfortunately, more than a few bandits and brigands have settled around here as well, taking advantage of the fact that the area has no local lord or authority to chase them off.