Several different sonic screwdrivers are shown sitting in a cup on the Doctor's desk, including models used by the Third, Fourth, and Eleventh Doctors.
The character of Bill was announced as the first openly gay character in the Doctor Who franchise. While this is true in the sense that she is attracted only to women, at least three of the Doctor's previous companions have expressed interest in multiple genders. Jack Harkness is best described as "pansexual", having had a number of lovers of various genders and species. Clara Oswald hinted at a relationship between herself and Jane Austen, and River Song (considered a companion in BBC lists) mentions seducing Cleopatra. In addition to this, the Doctor's allies Madame Vastra and Jenny are in a same-sex cross species marriage.
In the Doctor's office there are two busts. One of them is William Shakespeare and the other is Beethoven. Both are characters that the Doctor has met in his travels.
Framed photos of River Song and Susan Foreman (The Doctor's wife and granddaughter, respectively) are shown on The Doctor's desk. River was last seen in The Husbands of River Song (2015), which took place just before her death. Susan Foreman has not yet appeared in the new series (though some fans have speculated that the unnamed time lady featured in 'The End Of Time' might have been Susan; this has never been officially confirmed or denied).
The "star" in Heather's eye is a real condition called central heterochromia, where a person's iris has a ring inside of it which is a different color than the outer edge of the iris. The star shape of Heather's ring, however, is uniquely hers.