I am a Russian nobleman in the 19th century; how much would I want it to be known that my family came to prominence during the Mongol conquest? Would a Mongol or Tatar nobleman/noblewomen in the family tree be something to hide, or would I claim it as part of my heritage?

Generally speaking, no, you wouldn't be willing to hide it. Otherwise it depends on what exact level of "prominence" you imply. If you boast about your Mongol ancestors conquering their future slaves then it's possible you'll be treated as a really strange man, in the same way as Spanish nobleman telling everybody about his noble Arabian ancestors conquering his native peninsula, but not likely to be prosecuted.

In truth, there were many Tatar lineages throughout Russian history since Mongolian conquest, and they were fully integrated into society after pledging the allegiance with Russian tsars or princes. Essentially, after the collapse of Mongol empire (and further - after dissolution of the Golden Horde), Moscow deemed it very important to recruit and give precedency to Tatar, Siberian and even Caucasian (i.e. from the Caucasian Mountains) noblemen in order to, first, strengthen their political position and be able to rely upon more numerous and loyal nobility and, second, use their visible grandeur position to attract former Mongolian remnants under Russian banner as either vassals or parts of the state.

Tatar nobility starts playing big role in Russian history since 15th century and by 16th century Tatars are very prominent on Russian political scene as both state governors and generals (for example, leader of the contingent of Mesherian Tatars, Sivinduk-murza, who died at the battle of Orsha, 1514). As for 19th century, then there is even a chance that your oriental ancestry will be considered to be something one can be proud of, as by late 19th century Russian cultural circles begin to glorify panmongolism and Russian "Asian origin", as can be seen in Vladimir Soloviev's "Panmongolism":

Pan Mongolism!

The name is monstrous

Yet it caresses my ear

As if filled with the portent

Of a grand divine fate.

Or in Alexander Block's "Scythians":

Millions are you – and hosts, yea hosts, are we,And we shall fight if war you want, take heed.Yes, we are Scythians – leafs of the Asian tree,Our slanted eyes are bright aglow with greed.

Edit: Sivinduk-murza was captured at Orsha, not killed, my bad.