Do lysosomes play any role in cell division?
It is actually not completely correct to include it as a fact in textbooks. The main cause of it is that there is not much research on this topic. See this article for reference. According to Allison and Mallucci (1964), during cell division, lysosomes either released an unspecified activator or some hydrolase that inactivated a repressor. Apart from this, Allison and Paton (1965) showed that treatment of human embryonic lung cells in tissue culture with photosensitizing agents allowed high frequency of chromosome breaks on irradiating the cells with light from a high intensity tungsten source. They claimed that these effects were observed only at lysosomal sites, from which they concluded that a cytoplasmic event was giving rise to structural alteration in chromosomes. This event was suggested to be release of deoxyribonuclease from the lysosomes. However, Chevremont et al, 1959 showed different results through their experiments. They grew mammalian cells in tissue culture in the presence of acid deoxyribonuclease. This treatment inhibited cell division, but allowed DNA synthesis to proceed and did not appear to alter the nuclear DNA. Hence, the contribution of lysosomes in cell division is not yet completely established and still comes under debate.

So, it should not be considered a total false, but is also not reliable enough to be printed in textbooks. Overall, we don't yet know whether lysosomes do play a role or not.

References: 


TUATARA: VOLUME 18, ISSUE 1, JULY 1970
THE LYSOSOME
ALLISON AC, MALLUCCI L. LYSOSOMES IN DIVIDING CELLS, WITH SPECIAL REFERENCE TO LYMPHOCYTES. Lancet. 1964 Dec 26;2(7374):1371–1373.
Nature 227, 707 - 708 (15 August 1970); doi:10.1038/227707a0;
Chromosome Breakage by Deoxyribonuclease;
GILLIAN R. PATON &amp; A. C. ALLISON
CHEVREMONT M, CHEVREMONT-COMHAIRE S, BAECKELAND E. [Action of neutral and acid desoxyribonucleases on living somatic cells cultivated in vitro. III. Histoautoradiographic study after incorporation of tritium-labelled thymidine]. Arch Biol (Liege) 1959 Oct-Dec;70:833–849.