During one of my perpetual forages into Google looking for arcane information on Morphy, I came across this site about a Esplanade Avenue walking tour which offered up this information:

Denis-Morphy-Barnett Houses, nos. 600-06, were all constructed in 1834 by attorney Henry Raphael Denis on what was once three separate lots. The common-wall townhouses were basically all designed in the same style, although they appear as three distinct houses today, thanks to alterations through the years. The house at no. 600 has the particular distinction of having been owned by Alonzo Morphy, the father of famed chess master Paul Morphy. The gorgeous lacy two-level wraparound cast-iron balconies were added in the 1850s. The house at 604 originally possessed an entresol, or half-story, with Gothic arches, and the house at 606 was remodeled in 1859 in the Greek Revival style. It was once owned by Michel Bringier of l'Hermitage Plantation upriver from New Orleans.
 

Intriguing! I never realized that Judge Morphy owned a house on  Esplanade Avenue.

However, I had to corroborate this information, so I continued my search... and soon found myself on a site I've used many times in the past which did, indeed confirm this information saying that:

1. In 1835 Alonzo Morphy purchased the property from Henry Raphael Denis.

2. In 1836 Mr. and Mrs.Pierre Prodat bought the property from Alonzo Morphy.

3. In 1842 Jean Bautiste Sauvinet, an intimate and legal champion of the Baratarians who was living at the time at #124 Hospital St. (Gov. Nicholls), bought the property  from Pierre Prodat.

4. In 1860 it was used as a school for young children

I also learned from this site that on the Week of Sept 21st - 27th during some unspecified year,
600 Esplanade, with 8 bedrooms, 6 full bathrooms and 6791 sq. ft., sold for $1,050,000.

I learned that the North Club Hill Development, LLC, now located at 1322 Felicity St., was once located at 600 Esplanade Avenue.

And I found in The Times-Picayune, Sunday, June 27, 2004, listed as a new business, was Carte Blanche Inc., 600 Esplanade Ave., New Orleans, 70116 - owners:  Simon Blake,  Nadine Blake.

Further exploration turned up the following:

In 1809  Charles Louis Blache, Dame Felicite and Modeste Blache (Morphy's maternal grandmother) received  properties at 1205 Royal Street, 1211 Royal Street and 1215 Royal Street from Jean Joseph Blache. In 1818 the property  was sold to Martin Bourgeat (who, in turn, sold it to Jean Mercier and Dominque Bouligny in 1830)

In 1841 Joseph Le Carpentier bought the properties at 1235-1241 Royal Steet from Auguste O'Duhigg. In 1885 the propery housed a furniture store (from newspaper reports on a fire at that location).