This is the 2062nd most frequent Turkish word.
yenisi
"The new one."
Here, 'yenisini' is the accusative form of 'yenisi', which means 'the new one' (specifically referring to the new pen in this context).
Eski kalemimi kaybettim, bu yüzden yenisini satın aldım.
I lost my old pen, so I bought a new one.
In this sentence, 'yenisine' is the dative form of 'yenisi', indicated as 'the new one' (speaking about the new apartment in this situation).
Arkadaşım eski dairesinden taşındı ve yenisine geçti.
My friend moved out of the old apartment and transferred to the new one.
In this example, 'yenisini' is the accusative form of 'yenisi', referring again to 'the new one' (relative to shoes considered here).