  We explore the potential of the electric quadrupole transitions
$7s\,^2S_{1/2}$ - $6d\,^2D_{3/2}$, $6d\,^2D_{5/2}$ in radium isotopes as
single-ion optical frequency standards. The frequency shifts of the clock
transitions due to external fields and the corresponding uncertainties are
calculated. Several competitive $^A$Ra$^+$ candidates with $A=$ 223 - 229 are
identified. In particular, we show that the transition
$7s\,^2S_{1/2}\,(F=2,m_F=0)$ - $6d\,^2D_{3/2}\,(F=0,m_F=0)$ at 828 nm in
$^{223}$Ra$^+$, with no linear Zeeman and electric quadrupole shifts, stands
out as a relatively simple case, which could be exploited as a compact, robust,
and low-cost atomic clock operating at a fractional frequency uncertainty of
$10^{-17}$. With more experimental effort, the $^{223,225,226}$Ra$^+$ clocks
could be pushed to a projected performance reaching the $10^{-18}$ level.
