What type of mutation causes Progeria?
Hutchinson–Gilford progeria syndrome is almost always due to to a de novo mutation (i.e. not an inherited mutation) in the lamin A gene (LMNA). The mutation responsible is a C-to-T substitution at position 1824. Remarkably this doesn't change the encoded amino acid but rather creates a new splice donor site in the RNA transcript. When this splice site is used it results in the deletion of information corresponding to about 50 amino acids internal to the protein and near to the C terminus.

During biogenesis of normal lamin A a C-terminal -CAAX motif is recognised as a signal for isoprenylation, in this case farnesylation. The added lipid group acts as a C-terminal membrane anchor. However this anchor is subsequently removed when the protein is proteolytically cleaved near the C-terminus, removing 15 amino acids including the C-terminal cysteine (by this point the three C-terminal residues have been trimmed away). The cleavage site for this final modification is missing in HGPS lamin A so the farnesyl group cannot be removed. The accumulation of unprocessed lamin A causes the observed defects in the nuclear membrane.

Eriksson et al. (2003) Recurrent de novo point mutations in lamin A cause Hutchinson-Gilford progeria syndrome. Nature 423: 293-298 

Capell et al. (2005) Inhibiting farnesylation of progerin prevents the characteristic nuclear blebbing of Hutchinson-Gilford progeria syndrome. Proc. Natl. Acad. Sci. USA 102: 12879-12884