All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
It is forbidden for a woman to travel without a Mahram (i.e. non-marriageable relative) or a husband, this is either to spend her vacation or for other reasons, unless it is an obligatory travel like travelling to perform the obligatory Hajj, as some scholars permitted it, when with a safe company. For more details and benefit on this, please refer to Fatwa 85170.
As regards her spending the night in the house of her paternal aunt, then there is no harm on this, whether or not there is a sound religious reason for this, and whether or not her cousin is present there, provided this travel is with the permission of her husband or her guardian, and provided the place where she spends the night is safe and she does not stay in seclusion in it with a marriageable man.
Allaah Knows best.