What things that occur during Meiosis, help to increase the variety of genes in a gamete? How do they work?

Crossing-over mixes up segments of homologous chromosomes, resulting in a variety of alleles in the gametes.

There are two principal mechanisms by which genetic variation is enhanced in meiosis:
1) crossing over, the exchange of segments of DNA on the arms of maternal and paternal homologues during the pachytene substage of meiotic prophase 1, results in gametes with different combinations of alleles.
2) independent assortment, which is simply the observation that when maternal and paternal homologues line up on the metaphase equatorial plane in meiosis 1, they do so in a random fashion, i.e. the set of homologues on either side of the plane is likely to be a combination of maternal and paternal homologues, but the precise combination may vary from one gamete to the next.  This alone gives rise to 2^23 > 8,000,000 different gametes in any individual human, crossing over notwithstanding.