Can members pay their dues early? Basically before the system has them as expired.
If they renew early, it should just extend their membership. (e.g. if it is an annual membership and they have 2 months left when they renew, they should have 14 months on their membership afterwards).
If you are using memberships with fixed dates (e.g. calendar year memberships), then there is a setting "Fixed Period Rollover Day" that will control at which point a membership for the following period can be added.