First off take a look at this.
geology-mt-rushmore

A migmatite is a type of metamorphic rock that was once a mixture of a silicate liquid and a restite of minerals. Migmatite means mixed rock. Most migmatites form when a solid metamorphic rock gets heated due to intrusion of magma (often granitic magma). The granite magma slowly cools and the heat flows into the surround country rock; now, the heat flows in and the metamorphic rock, initially ALL SOLID, heats up and begins to melt! We call this process PARTIAL melting. The liquid that forms is light in color whereas the residue -the minerals that remain in solid state and do not melt- are generally dark in color. We call these two parts of the migmatite (mixed rock) the LEUCOSOME (light part) and melanosome (dark part).

Eventually, the granite magma solidifies and the migmatite cools and preserves the migmatite, layering of white bands within a darker matrix, the so called migmatite.

As far as Gold goes?
Well, while the granite is cooling and heat is flowing into the rock surrounding the granite magma, sometimes, if small amounts of water are present, the water gets heated up and can dissolve small amounts of Au. The Au is carried in chloride complexes in solution like AuCl3, etc.

The solutions move up, cool, and the gold is precipitated from the watery solutions, often with other minerals like quartz. This forms a disseminated Au deposit where the rock, with TINY flakes of Au is mined, crushed, and the Au separated.

A migmatite is a rock body that has been heated so much that some of the minerals in it have started to melt and segregate from other minerals with a higher melting temperature. Migmatites are often found at the base of crustal mountain blocks, where they have been heated by the pressure of the overlying mountains. The granite of Mt. Rushmore into which the faces of former U.S. presidents has been carved formed differently -- it is a batholith, from the cooling and crystallization of a large body of magma in the crust. There is a currently a batholith forming beneath the Cascade Volcanoes of Washington and Oregon.

Gold in the Black Hills include both placer deposits (small gold particles interspersed in stream sediments along the rivers draining the mountains) and primary ore bodies found in the bedrock. The biggest sources of gold in the area were the ore bodies themselves, including the 'Homestake' deposit. It has produced over 1.25 million kg of gold since it was first discovered in 1876. The homestake gold is found mainly in Precambrian (>540 million-year-old) metamorphic rocks. There have been several other stages of gold mineralization in the area, most recently during magamatism in the Tertiary (~50-60 million years ago). But the biggest gold deposits formed as a result of hot, mineral- rich waters moving through the metamorphic rocks of the Homestake formation. Gold-bearing minerals were deposited in large quantities along the hydrothermal pathways.

Migmatite rock is a particular metamorphic "facies." Assigning rock types to facies are a way of categorizing the rocks according to their specific patterning, which is the result of how they initially formed. Usually temperature and pressure are used to describe how the rocks form. What this means is that progressively increasing temperature and pressure results in a "series" of metamorphic facies, and this is a way to describe the metamorphic process. So migmatite rock refers to a particular type of rock in that series: rock with intricate folding patterns. The folded patterns occur because migmatite forms when there is partial melting of pre-existing rock.

This is related to the Mt. Rushmore granites, and I think many granites in general. Granites are igneous rocks that form from magma, which is melted rock. Migmatite rock lies at the interface between metamorphic and igneous rock. What I mean by this is that migmatites are metamorphic rocks formed via melting of rock -- but melted rock is magma -- and igneous rock comes from magma. So in a way, migmatite is a "stepping stone" on the path to granite.

Sorry, but I don't know how the Black Hills Gold was deposited. Gold ore can typically form in a variety of ways, including via plate tectonic activity and sedimentary deposition.

I hope some of this helps!

I can't answer for Mt. Rushmore specifically or for the Black Hills, but I can explain migmatite: minerals rich in iron and magnesium such as olivine and pyroxine have higher melting temperatures than minerals rich in silicon and oxygen like quartz and orthoclase. As a result, it is possible to heat a rock to the point where these lighter minerals melt, forming lenses of granite-like material, but the heavier minerals remain solid, forming gneiss-like bands. This composite rock is called a migmatite.