Per title, I must emphasize that it shall be on the near side. Antoniadi (lunar crater) and Aitken basin are all on the far side. Preferrably it is also wide and large like the Oceanus Procellarum...
Looks like it's Hercules G, which is a crater nested inside the larger crater of Hercules. It's about 10 km across, and Hercules is about 45 km across.
You can look on the LROC Quickmap for stuff like this. I created a link for you to the right area, and set up the layers to show topography. Explore the menus on the left for more stuff, it has all kinds of data. To see what the colors in the topography map mean, you can check the palette used for the layer. This is the WAC DTM Color Map, and there is a link to the palette at the bottom of it's section. Or, to make it easy. it's here, and I'll put in an image of both the crater and the palette.
According to the colors, Hercules G is between 6210 and 6940 m below lunar 'sea level'. I didn't see anything lower, feel free to pan around in case there is something lower.
Following Kim Holder's advice to "pan around", there's also Clavius D (-58.7, -12.4), another embedded crater inside the larger Clavius.
It's one step lower in absolute elevation, in the -5400m to -6210m category, but the surrounding rim of Clavius is about 1.5km higher than Hercules, so possibly deeper depending on your definition.