Not much remained, really.

The problem is, that the Mongols did not settle in Europe nor conquered it completely (not counting Rus, of course. Entire Rus was a subject of Mongols). Their effort to seize Hungary in 1280’s was futile and the great incursion in 1240’s probably did not have the aim to conquer any land. From the European point of view, Mongols appeared from nowhere and then disappeared again.

In Easter Europe, the situation was different. Entire Rus was conquered by Mongols and its history since around 1250 on was shaped by its servitude. Russian princes had to achieve special permissions called jarligs from Mongol khans to rule in their realms and Mongols were a powerful ally for those princes who wished to gain more power in a power struggle within the Rus. It was not uncommon for Rus princes to rely on help of their Mongol overlords in order to secure their position.

On the other hand, the Mongol rule was not that harsh. Rus rulers had to pay taxes to the Mongol capital (or later the capital of the Golden Horde) and religious freedom was guaranteed. Orthodox Christians were not persecuted and the Church was not dissolved.

Moreover, Mongols did not even settle in Eastern Europe. Their empire was not ethnically homogenous and Mongols pertained only a small part of its population. They were the ruling elite, not ordinary people of the realm because of their low numbers. Steppe people are not generally as populous as their settled counterparts and the same applies to the Mongols. This prevented them from settling everywhere in the steppe.

The only artifacts I know from that era that have any relation to Mongols are the above mentioned jarligs. A few of them survived from the Middle Ages up to today. Another items could have been found but I am not aware of them.

For Europe, it was more important that Mongols united the steppe. Once again there was a single empire that rule the whole space between Dnieper and the Pacific Oceans and this allowed many Europeans to visit Eastern Asia. On of those people was Marco Polo, his father and uncle but they were not he only ones. Among European monks there were many who decided to travel across the steppe to spread Christianity. In this regard, Mongols helped to augment the European world.

And in the Eastern Europe, a new power was about to rise thanks to the Mongols conquest. Deep in the woods around river Moskva, a small fort was chosen by the posterity of Yaroslav Vsevolodovich as their new seat. Their politics relied heavily on the Mongols support and during centuries after the first Mongol incursion, they became the biggest power among Rus principalities. In the 15th century, princes of Moscow stopped paying the tribute to the Golden Horde and started their expansion all over the Eastern Europe and eventually to Asia.

These are the most important remnants of the Mongol rule in the steppe from the European perspective.